id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-15000 | Graph of Nonlinear Additive Function is Dense in the Plane | Let $f: \R \to \R$ be an additive function which is not linear.
Then the graph of $f$ is dense in the real number plane. | From Additive Function is Linear for Rational Factors:
:$\map f q = q \map f 1$
for all $q \in \Q$.
{{WLOG}}, let:
:$\map f q = q$
for all $q \in \Q$.
Since $f$ is not linear, let $\alpha \in \R \setminus \Q$ be such that:
:$\map f \alpha = \alpha + \delta$
for some $\delta \ne 0$.
Consider an arbitrary nonempty circle... | Let $f: \R \to \R$ be an [[Definition:Additive Function (Algebra)|additive function]] which is not linear.
Then the [[Definition:Graph of Mapping|graph]] of $f$ is [[Definition:Everywhere Dense|dense]] in the [[Definition:Real Number Plane|real number plane]]. | From [[Additive Function is Linear for Rational Factors]]:
:$\map f q = q \map f 1$
for all $q \in \Q$.
{{WLOG}}, let:
:$\map f q = q$
for all $q \in \Q$.
Since $f$ is not linear, let $\alpha \in \R \setminus \Q$ be such that:
:$\map f \alpha = \alpha + \delta$
for some $\delta \ne 0$.
Consider an arbitrary nonempt... | Graph of Nonlinear Additive Function is Dense in the Plane | https://proofwiki.org/wiki/Graph_of_Nonlinear_Additive_Function_is_Dense_in_the_Plane | https://proofwiki.org/wiki/Graph_of_Nonlinear_Additive_Function_is_Dense_in_the_Plane | [
"Additive Functions"
] | [
"Definition:Additive Function (Algebra)",
"Definition:Graph of Mapping",
"Definition:Everywhere Dense",
"Definition:Real Number Plane"
] | [
"Additive Function is Linear for Rational Factors",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Topology",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Topology",
"Additive Function is Linear for Rational Factors",
"Additive Function is Linear for Rational Factors"
] |
proofwiki-15001 | Order of Elements in Quaternion Group | Let $Q = \Dic 2$ be the quaternion group, whose group presentation is given by:
:$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$
Then $\Dic 2$ has:
:$1$ element of order $2$
and:
:$6$ elements of order $4$. | From Identity is Only Group Element of Order 1, the identity element $e$ , and only $e$, is of order $1$.
From here, we inspect the Cayley table:
{{:Quaternion Group/Cayley Table}}
It is immediately seen that:
:$\paren {a^2}^2 = e$
and so by definition $a^2$ is of order $2$.
As can be seen from inspection of the main d... | Let $Q = \Dic 2$ be the [[Definition:Quaternion Group|quaternion group]], whose [[Definition:Group Presentation|group presentation]] is given by:
:$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$
Then $\Dic 2$ has:
:$1$ [[Definition:Element|element]] of [[Definition:Order of Group Element|order]] $2$
and:
:$6$ [... | From [[Identity is Only Group Element of Order 1]], the [[Definition:Identity Element|identity element]] $e$ , and only $e$, is of [[Definition:Order of Group Element|order $1$]].
From here, we inspect the [[Quaternion Group/Cayley Table|Cayley table]]:
{{:Quaternion Group/Cayley Table}}
It is immediately seen that:
... | Order of Elements in Quaternion Group | https://proofwiki.org/wiki/Order_of_Elements_in_Quaternion_Group | https://proofwiki.org/wiki/Order_of_Elements_in_Quaternion_Group | [
"Quaternion Group"
] | [
"Definition:Dicyclic Group/Quaternion Group",
"Definition:Group Presentation",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Element",
"Definition:Order of Group Element"
] | [
"Identity is Only Group Element of Order 1",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Order of Group Element",
"Quaternion Group/Cayley Table",
"Definition:Order of Group Element",
"Definition:Matrix/Diagonal/Main",
"Definition:Element",
"Definition:Order of Group Eleme... |
proofwiki-15002 | Multiplicative Group of Reduced Residues Modulo 7 is Cyclic | Let $\struct {\Z'_7, \times_7}$ denote the multiplicative group of reduced residues modulo $7$.
Then $\struct {\Z'_7, \times_7}$ is cyclic. | From Reduced Residue System under Multiplication forms Abelian Group it is noted that $\struct {\Z'_7, \times_7}$ is a group.
It remains to be shown that $\struct {\Z'_7, \times_7}$ is cyclic.
It will be demonstrated that:
:$\gen {\eqclass 3 7} = \struct {\Z'_7, \times_7}$
That is, that $\eqclass 3 7$ is a generator of... | Let $\struct {\Z'_7, \times_7}$ denote the [[Multiplicative Group of Reduced Residues Modulo 7|multiplicative group of reduced residues modulo $7$]].
Then $\struct {\Z'_7, \times_7}$ is [[Definition:Cyclic Group|cyclic]]. | From [[Reduced Residue System under Multiplication forms Abelian Group]] it is noted that $\struct {\Z'_7, \times_7}$ is a [[Definition:Group|group]].
It remains to be shown that $\struct {\Z'_7, \times_7}$ is [[Definition:Cyclic Group|cyclic]].
It will be demonstrated that:
:$\gen {\eqclass 3 7} = \struct {\Z'_7, \t... | Multiplicative Group of Reduced Residues Modulo 7 is Cyclic | https://proofwiki.org/wiki/Multiplicative_Group_of_Reduced_Residues_Modulo_7_is_Cyclic | https://proofwiki.org/wiki/Multiplicative_Group_of_Reduced_Residues_Modulo_7_is_Cyclic | [
"Multiplicative Groups of Reduced Residues",
"Multiplicative Group of Reduced Residues Modulo 7",
"Examples of Cyclic Groups"
] | [
"Multiplicative Group of Reduced Residues/Examples/Modulo 7",
"Definition:Cyclic Group"
] | [
"Reduced Residue System under Multiplication forms Abelian Group",
"Definition:Group",
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Power of Element/Group",
"Definition:Element",
"Definition:Cyclic Group"
] |
proofwiki-15003 | Reduced Residues Modulo 5 under Multiplication form Cyclic Group | Let $\struct {\Z'_5, \times_5}$ denote the multiplicative group of reduced residues modulo $5$.
Then $\struct {\Z'_5, \times_5}$ is cyclic. | From Reduced Residue System under Multiplication forms Abelian Group it is noted that $\struct {\Z'_5, \times_5}$ is a group.
It remains to be shown that $\struct {\Z'_5, \times_5}$ is cyclic.
It will be demonstrated that:
:$\gen {\eqclass 2 5} = \struct {\Z'_5, \times_5}$
That is, that $\eqclass 2 5$ is a generator of... | Let $\struct {\Z'_5, \times_5}$ denote the [[Multiplicative Group of Reduced Residues Modulo 5|multiplicative group of reduced residues modulo $5$]].
Then $\struct {\Z'_5, \times_5}$ is [[Definition:Cyclic Group|cyclic]]. | From [[Reduced Residue System under Multiplication forms Abelian Group]] it is noted that $\struct {\Z'_5, \times_5}$ is a [[Definition:Group|group]].
It remains to be shown that $\struct {\Z'_5, \times_5}$ is [[Definition:Cyclic Group|cyclic]].
It will be demonstrated that:
:$\gen {\eqclass 2 5} = \struct {\Z'_5, \t... | Reduced Residues Modulo 5 under Multiplication form Cyclic Group | https://proofwiki.org/wiki/Reduced_Residues_Modulo_5_under_Multiplication_form_Cyclic_Group | https://proofwiki.org/wiki/Reduced_Residues_Modulo_5_under_Multiplication_form_Cyclic_Group | [
"Multiplicative Groups of Reduced Residues",
"Multiplicative Group of Reduced Residues Modulo 5",
"Examples of Cyclic Groups",
"Cyclic Group of Order 4"
] | [
"Multiplicative Group of Reduced Residues/Examples/Modulo 5",
"Definition:Cyclic Group"
] | [
"Reduced Residue System under Multiplication forms Abelian Group",
"Definition:Group",
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Power of Element/Group",
"Definition:Element",
"Definition:Cyclic Group",
"C... |
proofwiki-15004 | Number of Generators of Cyclic Group whose Order is Power of 2 | Let $G$ be a finite cyclic group.
Let the order of $G$ be $2^k$ for some $k \in \Z_{>0}$.
Then $G$ has $2^{n - 1}$ distinct generators. | From Finite Cyclic Group has Euler Phi Generators, $G$ has $\map \phi {2^n}$ generators.
The result follows from {{Corollary|Euler Phi Function of Prime Power}}.
{{qed}} | Let $G$ be a [[Definition:Finite Group|finite]] [[Definition:Cyclic Group|cyclic group]].
Let the [[Definition:Order of Structure|order]] of $G$ be $2^k$ for some $k \in \Z_{>0}$.
Then $G$ has $2^{n - 1}$ [[Definition:Distinct|distinct]] [[Definition:Generator of Group|generators]]. | From [[Finite Cyclic Group has Euler Phi Generators]], $G$ has $\map \phi {2^n}$ [[Definition:Generator of Group|generators]].
The result follows from {{Corollary|Euler Phi Function of Prime Power}}.
{{qed}} | Number of Generators of Cyclic Group whose Order is Power of 2 | https://proofwiki.org/wiki/Number_of_Generators_of_Cyclic_Group_whose_Order_is_Power_of_2 | https://proofwiki.org/wiki/Number_of_Generators_of_Cyclic_Group_whose_Order_is_Power_of_2 | [
"Cyclic Groups",
"Generators of Groups"
] | [
"Definition:Finite Group",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Distinct",
"Definition:Generator of Group"
] | [
"Finite Cyclic Group has Euler Phi Generators",
"Definition:Generator of Group"
] |
proofwiki-15005 | Equivalent Statements for Congruence Modulo Subgroup/Left | Let $x \equiv^l y \pmod H$ denote that $x$ is left congruent modulo $H$ to $y$.
Then the following statements are equivalent:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \equiv^l
| r = y \pmod H
}}
{{eqn | n = 2
| l = x^{-1} y
| o = \in
| r = H
}}
{{eqn | n = 3
| q = \exists h \i... | {{begin-eqn}}
{{eqn | l = x
| o = \equiv^l
| r = y \pmod H
}}
{{eqn | ll= \leadstoandfrom
| l = x^{-1} y
| o = \in
| r = H
| c = {{Defof|Left Congruence Modulo Subgroup}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists h \in H
| l = x^{-1} y
| r = h
| c = {{Defof... | Let $x \equiv^l y \pmod H$ denote that $x$ is [[Definition:Left Congruence Modulo Subgroup|left congruent modulo $H$]] to $y$.
Then the following statements are [[Definition:Logical Equivalence|equivalent]]:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \equiv^l
| r = y \pmod H
}}
{{eqn | n = 2
| ... | {{begin-eqn}}
{{eqn | l = x
| o = \equiv^l
| r = y \pmod H
}}
{{eqn | ll= \leadstoandfrom
| l = x^{-1} y
| o = \in
| r = H
| c = {{Defof|Left Congruence Modulo Subgroup}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists h \in H
| l = x^{-1} y
| r = h
| c = {{Defof... | Equivalent Statements for Congruence Modulo Subgroup/Left | https://proofwiki.org/wiki/Equivalent_Statements_for_Congruence_Modulo_Subgroup/Left | https://proofwiki.org/wiki/Equivalent_Statements_for_Congruence_Modulo_Subgroup/Left | [
"Congruence Modulo Subgroup"
] | [
"Definition:Congruence Modulo Subgroup/Left Congruence",
"Definition:Logical Equivalence"
] | [
"Division Laws for Groups"
] |
proofwiki-15006 | Equivalent Statements for Congruence Modulo Subgroup/Right | Let $x \equiv^r y \pmod H$ denote that $x$ is right congruent modulo $H$ to $y$.
Then the following statements are equivalent:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \equiv^r
| r = y \pmod H
}}
{{eqn | n = 2
| l = x y^{-1}
| o = \in
| r = H
}}
{{eqn | n = 3
| q = \exists h \... | {{begin-eqn}}
{{eqn | l = x
| o = \equiv^r
| r = y \pmod H
}}
{{eqn | ll= \leadstoandfrom
| l = x y^{-1}
| o = \in
| r = H
| c = {{Defof|Right Congruence Modulo Subgroup}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists h \in H
| l = x y^{-1}
| r = h
| c = {{Defo... | Let $x \equiv^r y \pmod H$ denote that $x$ is [[Definition:Right Congruence Modulo Subgroup|right congruent modulo $H$]] to $y$.
Then the following statements are [[Definition:Logical Equivalence|equivalent]]:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \equiv^r
| r = y \pmod H
}}
{{eqn | n = 2
... | {{begin-eqn}}
{{eqn | l = x
| o = \equiv^r
| r = y \pmod H
}}
{{eqn | ll= \leadstoandfrom
| l = x y^{-1}
| o = \in
| r = H
| c = {{Defof|Right Congruence Modulo Subgroup}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists h \in H
| l = x y^{-1}
| r = h
| c = {{Defo... | Equivalent Statements for Congruence Modulo Subgroup/Right | https://proofwiki.org/wiki/Equivalent_Statements_for_Congruence_Modulo_Subgroup/Right | https://proofwiki.org/wiki/Equivalent_Statements_for_Congruence_Modulo_Subgroup/Right | [
"Congruence Modulo Subgroup"
] | [
"Definition:Congruence Modulo Subgroup/Right Congruence",
"Definition:Logical Equivalence"
] | [
"Division Laws for Groups"
] |
proofwiki-15007 | Element of Group is in its own Coset/Left | Let:
: $x H$ be the left coset of $x$ modulo $H$.
Then:
: $x \in x H$ | Let $e$ be the identity of $G$.
Then:
{{begin-eqn}}
{{eqn | l = e
| o = \in
| r = H
| c = Identity of Subgroup
}}
{{eqn | l = x
| r = x e
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \leadsto
| q = \exists h \in H
| l = x
| r = x h
| c = Existential Generalisatio... | Let:
: $x H$ be the [[Definition:Left Coset|left coset]] of $x$ modulo $H$.
Then:
: $x \in x H$ | Let $e$ be the [[Definition:Identity Element|identity]] of $G$.
Then:
{{begin-eqn}}
{{eqn | l = e
| o = \in
| r = H
| c = [[Identity of Subgroup]]
}}
{{eqn | l = x
| r = x e
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \leadsto
| q = \exists h \in H
| l = x
| r = x h
... | Element of Group is in its own Coset/Left | https://proofwiki.org/wiki/Element_of_Group_is_in_its_own_Coset/Left | https://proofwiki.org/wiki/Element_of_Group_is_in_its_own_Coset/Left | [
"Element of Group is in its own Coset"
] | [
"Definition:Coset/Left Coset"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Identity of Subgroup",
"Existential Generalisation"
] |
proofwiki-15008 | Element of Group is in its own Coset/Right | Let:
: $H x$ be the right coset of $x$ modulo $H$.
Then:
: $x \in H x$ | Let $e$ be the identity of $G$.
Then:
{{begin-eqn}}
{{eqn | l = e
| o = \in
| r = H
| c = Identity of Subgroup
}}
{{eqn | l = x
| r = e x
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \leadsto
| q = \exists h \in H
| l = x
| r = h x
| c = Existential Generalisatio... | Let:
: $H x$ be the [[Definition:Right Coset|right coset]] of $x$ modulo $H$.
Then:
: $x \in H x$ | Let $e$ be the [[Definition:Identity Element|identity]] of $G$.
Then:
{{begin-eqn}}
{{eqn | l = e
| o = \in
| r = H
| c = [[Identity of Subgroup]]
}}
{{eqn | l = x
| r = e x
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \leadsto
| q = \exists h \in H
| l = x
| r = h x
... | Element of Group is in its own Coset/Right | https://proofwiki.org/wiki/Element_of_Group_is_in_its_own_Coset/Right | https://proofwiki.org/wiki/Element_of_Group_is_in_its_own_Coset/Right | [
"Element of Group is in its own Coset"
] | [
"Definition:Coset/Right Coset"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Identity of Subgroup",
"Existential Generalisation"
] |
proofwiki-15009 | Element of Group is in Unique Coset of Subgroup/Left | There exists a exactly one left coset of $H$ containing $x$, that is: $x H$ | Follows directly from:
* Left Congruence Modulo Subgroup is Equivalence Relation
* Element in its own Equivalence Class.
{{qed}} | There exists a [[Definition:Unique|exactly one]] [[Definition:Left Coset|left coset]] of $H$ containing $x$, that is: $x H$ | Follows directly from:
* [[Left Congruence Modulo Subgroup is Equivalence Relation]]
* [[Element in its own Equivalence Class]].
{{qed}} | Element of Group is in Unique Coset of Subgroup/Left | https://proofwiki.org/wiki/Element_of_Group_is_in_Unique_Coset_of_Subgroup/Left | https://proofwiki.org/wiki/Element_of_Group_is_in_Unique_Coset_of_Subgroup/Left | [
"Element of Group is in Unique Coset of Subgroup"
] | [
"Definition:Unique",
"Definition:Coset/Left Coset"
] | [
"Left Congruence Modulo Subgroup is Equivalence Relation",
"Element in its own Equivalence Class"
] |
proofwiki-15010 | Element of Group is in Unique Coset of Subgroup/Right | There exists a exactly one right coset of $H$ containing $x$, that is: $H x$ | Follows directly from:
* Right Congruence Modulo Subgroup is Equivalence Relation
* Element in its own Equivalence Class.
{{qed}} | There exists a [[Definition:Unique|exactly one]] [[Definition:Right Coset|right coset]] of $H$ containing $x$, that is: $H x$ | Follows directly from:
* [[Right Congruence Modulo Subgroup is Equivalence Relation]]
* [[Element in its own Equivalence Class]].
{{qed}} | Element of Group is in Unique Coset of Subgroup/Right | https://proofwiki.org/wiki/Element_of_Group_is_in_Unique_Coset_of_Subgroup/Right | https://proofwiki.org/wiki/Element_of_Group_is_in_Unique_Coset_of_Subgroup/Right | [
"Element of Group is in Unique Coset of Subgroup"
] | [
"Definition:Unique",
"Definition:Coset/Right Coset"
] | [
"Right Congruence Modulo Subgroup is Equivalence Relation",
"Element in its own Equivalence Class"
] |
proofwiki-15011 | Equivalence of Definitions of Infinite Order Element | {{TFAE|def = Infinite Order Element}}
Let $G$ be a group whose identity is $e_G$.
Let $x \in G$ be an element of $G$. | === $(1)$ implies $(2)$ ===
Let $x$ be an infinite order element of $G$ by definition 1.
Then by definition there exists no $k \in \Z_{>0}$ such that $x^k = e_G$.
{{AimForCont}} not all $x^r$ are distinct for all $r \in \Z_{>0}$.
Then $x^m = x^n$ for some $m, n \in \Z$ where $m \ne n$.
{{WLOG}}, let $m > n$.
Then:
: $x... | {{TFAE|def = Infinite Order Element}}
Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e_G$.
Let $x \in G$ be an [[Definition:Element|element]] of $G$. | === $(1)$ implies $(2)$ ===
Let $x$ be an [[Definition:Order of Group Element/Infinite/Definition 1|infinite order element of $G$ by definition 1]].
Then by definition there exists no $k \in \Z_{>0}$ such that $x^k = e_G$.
{{AimForCont}} not all $x^r$ are [[Definition:Distinct Elements|distinct]] for all $r \in \Z_{... | Equivalence of Definitions of Infinite Order Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Infinite_Order_Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Infinite_Order_Element | [
"Order of Group Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element"
] | [
"Definition:Order of Group Element/Infinite/Definition 1",
"Definition:Distinct/Plural",
"Definition:Contradiction",
"Definition:Distinct",
"Definition:Order of Group Element/Infinite/Definition 2",
"Definition:Order of Group Element/Infinite/Definition 2",
"Definition:Distinct/Plural",
"Definition:Co... |
proofwiki-15012 | Equivalence of Definitions of Finite Order Element | {{TFAE|def = Finite Order Element}}
Let $G$ be a group whose identity is $e_G$.
Let $x \in G$ be an element of $G$. | === $(1)$ implies $(2)$ ===
Let $x$ be a finite order element of $G$ by definition 1.
Then by definition there exists $k \in \Z_{>0}$ such that $x^k = e_G$.
Consider some $m, n \in \Z_{>0}$ such that $m = n + k$.
{{begin-eqn}}
{{eqn | l = x^m
| r = x^{n + k}
| c = {{hypothesis}}
}}
{{eqn | r = x^n x^k
... | {{TFAE|def = Finite Order Element}}
Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e_G$.
Let $x \in G$ be an [[Definition:Element|element]] of $G$. | === $(1)$ implies $(2)$ ===
Let $x$ be a [[Definition:Order of Group Element/Finite/Definition 1|finite order element of $G$ by definition 1]].
Then by definition there exists $k \in \Z_{>0}$ such that $x^k = e_G$.
Consider some $m, n \in \Z_{>0}$ such that $m = n + k$.
{{begin-eqn}}
{{eqn | l = x^m
| r = x^{... | Equivalence of Definitions of Finite Order Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Finite_Order_Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Finite_Order_Element | [
"Order of Group Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element"
] | [
"Definition:Order of Group Element/Finite/Definition 1",
"Powers of Group Elements/Sum of Indices",
"Definition:Order of Group Element/Finite/Definition 2",
"Definition:Order of Group Element/Finite/Definition 2",
"Powers of Group Elements",
"Powers of Group Elements/Sum of Indices",
"Definition:Order o... |
proofwiki-15013 | Order of Element in Group equals its Order in Subgroup | Let $G$ be a group.
Let $H \le G$, where $\le$ denotes the property of being a subgroup.
Let $x \in H$.
Then the order of $x$ in $H$ equals the order of $x$ in $G$. | Let $\gen x$ be the subgroup of $G$ generated by $x$.
By definition, $\gen x \le G$.
All the elements of $\gen x$ are powers of $x$.
As $x \in H$ it follows by {{Group-axiom|0}} that all the powers of $x$ are elements of $H$.
That is:
:$\gen x \le H$
By Order of Cyclic Group equals Order of Generator:
:$\order x = \ord... | Let $G$ be a [[Definition:Group|group]].
Let $H \le G$, where $\le$ denotes the property of being a [[Definition:Subgroup|subgroup]].
Let $x \in H$.
Then the [[Definition:Order of Group Element|order]] of $x$ in $H$ equals the [[Definition:Order of Group Element|order]] of $x$ in $G$. | Let $\gen x$ be the [[Definition:Generated Subgroup|subgroup of $G$ generated]] by $x$.
By definition, $\gen x \le G$.
All the [[Definition:Element|elements]] of $\gen x$ are [[Definition:Power of Group Element|powers]] of $x$.
As $x \in H$ it follows by {{Group-axiom|0}} that all the [[Definition:Power of Group Ele... | Order of Element in Group equals its Order in Subgroup | https://proofwiki.org/wiki/Order_of_Element_in_Group_equals_its_Order_in_Subgroup | https://proofwiki.org/wiki/Order_of_Element_in_Group_equals_its_Order_in_Subgroup | [
"Order of Group Elements",
"Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Order of Group Element",
"Definition:Order of Group Element"
] | [
"Definition:Generated Subgroup",
"Definition:Element",
"Definition:Power of Element/Group",
"Definition:Power of Element/Group",
"Definition:Element",
"Order of Cyclic Group equals Order of Generator",
"Definition:Cyclic Group"
] |
proofwiki-15014 | Order of Product of Commuting Group Elements of Coprime Order is Product of Orders | Let $G$ be a group.
Let $g_1, g_2 \in G$ be commuting elements such that:
{{begin-eqn}}
{{eqn | l = \order {g_1}
| r = n_1
}}
{{eqn | l = \order {g_2}
| r = n_2
}}
{{end-eqn}}
where $\order {g_1}$ denotes the order of $g_1$ in $G$.
Let $n_1$ and $n_2$ be coprime.
Then:
:$\order {g_1 g_2} = n_1 n_2$ | Let $g_1 g_2 = g_2 g_1$.
We have:
:$\paren {g_1 g_2}^{n_1 n_2} = e$
Thus:
:$\order {g_1 g_2} \le n_1 n_2$
Suppose $\order {g_1 g_2}^r = e$.
Then:
{{begin-eqn}}
{{eqn | l = {g_1}^r
| r = {g_2}^{-r}
| c =
}}
{{eqn | o = \in
| r = \gen {g_1} \cap \gen {g_2}
| c =
}}
{{eqn | ll= \leadsto
| l... | Let $G$ be a [[Definition:Group|group]].
Let $g_1, g_2 \in G$ be [[Definition:Commuting Elements|commuting elements]] such that:
{{begin-eqn}}
{{eqn | l = \order {g_1}
| r = n_1
}}
{{eqn | l = \order {g_2}
| r = n_2
}}
{{end-eqn}}
where $\order {g_1}$ denotes the [[Definition:Order of Group Element|order... | Let $g_1 g_2 = g_2 g_1$.
We have:
:$\paren {g_1 g_2}^{n_1 n_2} = e$
Thus:
:$\order {g_1 g_2} \le n_1 n_2$
Suppose $\order {g_1 g_2}^r = e$.
Then:
{{begin-eqn}}
{{eqn | l = {g_1}^r
| r = {g_2}^{-r}
| c =
}}
{{eqn | o = \in
| r = \gen {g_1} \cap \gen {g_2}
| c =
}}
{{eqn | ll= \leadsto
... | Order of Product of Commuting Group Elements of Coprime Order is Product of Orders | https://proofwiki.org/wiki/Order_of_Product_of_Commuting_Group_Elements_of_Coprime_Order_is_Product_of_Orders | https://proofwiki.org/wiki/Order_of_Product_of_Commuting_Group_Elements_of_Coprime_Order_is_Product_of_Orders | [
"Order of Group Elements"
] | [
"Definition:Group",
"Definition:Commutative/Elements",
"Definition:Order of Group Element",
"Definition:Coprime/Integers"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-15015 | Right Cosets are Equal iff Left Cosets by Inverse are Equal | Let $G$ be a group whose identity is $e$.
Let $H$ be a subgroup of $G$.
Let $g_1, g_2 \in G$.
Then:
:$H g_1 = H g_2 \iff {g_1}^{-1} H = {g_2}^{-1} H$
where:
:${g_1}^{-1}$ and ${g_2}^{-1}$ denote the inverses of $g_1$ and $g_2$ in $G$
:$H g_1$ and $H g_2$ denote the right cosets of $H$ by $g_1$ and $g_2$ respectively
:$... | {{begin-eqn}}
{{eqn | l = H g_1
| r = H g_2
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = g_1 {g_2}^{-1}
| o = \in
| r = H
| c = Right Cosets are Equal iff Product with Inverse in Subgroup
}}
{{eqn | ll= \leadstoandfrom
| l = \paren { {g_1}^{-1} }^{-1} {g_2}^{-1}
| o = \in
... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $g_1, g_2 \in G$.
Then:
:$H g_1 = H g_2 \iff {g_1}^{-1} H = {g_2}^{-1} H$
where:
:${g_1}^{-1}$ and ${g_2}^{-1}$ denote the [[Definition:Inverse Element|inverses]]... | {{begin-eqn}}
{{eqn | l = H g_1
| r = H g_2
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = g_1 {g_2}^{-1}
| o = \in
| r = H
| c = [[Right Cosets are Equal iff Product with Inverse in Subgroup]]
}}
{{eqn | ll= \leadstoandfrom
| l = \paren { {g_1}^{-1} }^{-1} {g_2}^{-1}
| o = ... | Right Cosets are Equal iff Left Cosets by Inverse are Equal | https://proofwiki.org/wiki/Right_Cosets_are_Equal_iff_Left_Cosets_by_Inverse_are_Equal | https://proofwiki.org/wiki/Right_Cosets_are_Equal_iff_Left_Cosets_by_Inverse_are_Equal | [
"Cosets"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Coset/Right Coset",
"Definition:Coset/Left Coset"
] | [
"Right Cosets are Equal iff Product with Inverse in Subgroup",
"Inverse of Group Inverse",
"Left Cosets are Equal iff Product with Inverse in Subgroup"
] |
proofwiki-15016 | Subgroup of Subgroup with Prime Index | Let $\struct {G, \circ}$ be a group.
Let $H$ be a subgroup of $G$.
Let $K$ be a subgroup of $H$.
Let:
:$\index G K = p$
where:
:$p$ denotes a prime number
:$\index G K$ denotes the index of $K$ in $G$.
Then either:
:$H = K$
or:
:$H = G$ | From the Tower Law for Subgroups:
:$\index G K = \index G H \index H K$
As $\index G K = p$ is prime, either $\index G H = p$ or $\index H K = p$.
Thus either $\index G H = 1$ or $\index H K = 1$.
The result follows from Index is One iff Subgroup equals Group.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $K$ be a [[Definition:Subgroup|subgroup]] of $H$.
Let:
:$\index G K = p$
where:
:$p$ denotes a [[Definition:Prime Number|prime number]]
:$\index G K$ denotes the [[Definition:Index of Subgroup|index]]... | From the [[Tower Law for Subgroups]]:
:$\index G K = \index G H \index H K$
As $\index G K = p$ is [[Definition:Prime Number|prime]], either $\index G H = p$ or $\index H K = p$.
Thus either $\index G H = 1$ or $\index H K = 1$.
The result follows from [[Index is One iff Subgroup equals Group]].
{{qed}} | Subgroup of Subgroup with Prime Index | https://proofwiki.org/wiki/Subgroup_of_Subgroup_with_Prime_Index | https://proofwiki.org/wiki/Subgroup_of_Subgroup_with_Prime_Index | [
"Index of Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Prime Number",
"Definition:Index of Subgroup"
] | [
"Tower Law for Subgroups",
"Definition:Prime Number",
"Index is One iff Subgroup equals Group"
] |
proofwiki-15017 | Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2/Reverse Implication | :$\vdash \paren {\paren {p \lor q} \land \paren {p \lor r} } \implies \paren {p \lor \paren {q \land r} }$ | {{finish|Use formulation 1 + Modus Ponens}} | :$\vdash \paren {\paren {p \lor q} \land \paren {p \lor r} } \implies \paren {p \lor \paren {q \land r} }$ | {{finish|Use formulation 1 + Modus Ponens}} | Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2/Reverse Implication | https://proofwiki.org/wiki/Rule_of_Distribution/Disjunction_Distributes_over_Conjunction/Left_Distributive/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/Rule_of_Distribution/Disjunction_Distributes_over_Conjunction/Left_Distributive/Formulation_2/Reverse_Implication | [
"Rule of Distribution"
] | [] | [] |
proofwiki-15018 | Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations | Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations.
That is, for every WFF $\mathbf A$:
:$\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathscr H_2} \mathbf A$ | {{ProofWanted|This is going to be fun}} | [[Definition:Hilbert Proof System/Instance 2|Instance 2]] of the [[Definition:Hilbert Proof System|Hilbert proof systems]] is a [[Definition:Complete Proof System|complete proof system]] for [[Definition:Boolean Interpretation|boolean interpretations]].
That is, for every [[Definition:WFF of Propositional Logic|WFF]] ... | {{ProofWanted|This is going to be fun}} | Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations | https://proofwiki.org/wiki/Completeness_Theorem_for_Hilbert_Proof_System_Instance_2_and_Boolean_Interpretations | https://proofwiki.org/wiki/Completeness_Theorem_for_Hilbert_Proof_System_Instance_2_and_Boolean_Interpretations | [
"Completeness Theorem",
"Hilbert Proof System Instance 2",
"Named Theorems"
] | [
"Definition:Hilbert Proof System/Instance 2",
"Definition:Hilbert Proof System",
"Definition:Complete Proof System",
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] | [] |
proofwiki-15019 | Symmetry Group of Rectangle is Klein Four-Group | The symmetry group of the rectangle is the Klein $4$-group. | Comparing the Cayley tables of the symmetry group of the rectangle with the Klein $4$-group the isomorphism can be seen:
{{:Symmetry Group of Rectangle/Cayley Table}}
{{:Klein Four-Group/Cayley Table}}
Thus the required isomorphism $\phi$ can be set up as:
{{begin-eqn}}
{{eqn | l = \map \phi e
| r = e
}}
{{eqn |... | The [[Definition:Symmetry Group of Rectangle|symmetry group of the rectangle]] is the [[Definition:Klein Four-Group|Klein $4$-group]]. | Comparing the [[Definition:Cayley Table|Cayley tables]] of the [[Symmetry Group of Rectangle/Cayley Table|symmetry group of the rectangle]] with the [[Klein Four-Group/Cayley Table|Klein $4$-group]] the [[Definition:Group Isomorphism|isomorphism]] can be seen:
{{:Symmetry Group of Rectangle/Cayley Table}}
{{:Klein F... | Symmetry Group of Rectangle is Klein Four-Group | https://proofwiki.org/wiki/Symmetry_Group_of_Rectangle_is_Klein_Four-Group | https://proofwiki.org/wiki/Symmetry_Group_of_Rectangle_is_Klein_Four-Group | [
"Klein Four-Group"
] | [
"Definition:Symmetry Group of Rectangle",
"Definition:Klein Four-Group"
] | [
"Definition:Cayley Table",
"Symmetry Group of Rectangle/Cayley Table",
"Klein Four-Group/Cayley Table",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] |
proofwiki-15020 | Klein Four-Group as Subgroup of S4 | Let $G$ be the following subset of the symmetric group on $4$ letters $S_4$, expressed in two-row notation:
{{begin-eqn}}
{{eqn | l = e
| r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{bmatrix}
| c =
}}
{{eqn | l = a
| r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{bmatrix}
| ... | By inspection, the Cayley table is constructed:
:<nowiki>$\begin{array}{c|cccc}
& e & a & b & c \\
\hline
e & e & a & b & c \\
a & a & e & c & b \\
b & b & c & e & a \\
c & c & b & a & e \\
\end{array}$</nowiki>
Again by inspection this can be seen to be the same as the Cayley table for the Klein $4$-group.
{{qed}} | Let $G$ be the following [[Definition:Subset|subset]] of the [[Definition:Symmetric Group|symmetric group on $4$ letters]] $S_4$, expressed in [[Definition:Two-Row Notation|two-row notation]]:
{{begin-eqn}}
{{eqn | l = e
| r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{bmatrix}
| c =
}}
{{eqn | l... | By inspection, the [[Definition:Cayley Table|Cayley table]] is constructed:
:<nowiki>$\begin{array}{c|cccc}
& e & a & b & c \\
\hline
e & e & a & b & c \\
a & a & e & c & b \\
b & b & c & e & a \\
c & c & b & a & e \\
\end{array}$</nowiki>
Again by inspection this can be seen to be the same as the [[Klein Four-Grou... | Klein Four-Group as Subgroup of S4 | https://proofwiki.org/wiki/Klein_Four-Group_as_Subgroup_of_S4 | https://proofwiki.org/wiki/Klein_Four-Group_as_Subgroup_of_S4 | [
"Klein Four-Group"
] | [
"Definition:Subset",
"Definition:Symmetric Group",
"Definition:Permutation on n Letters/Two-Row Notation",
"Definition:Klein Four-Group"
] | [
"Definition:Cayley Table",
"Klein Four-Group/Cayley Table"
] |
proofwiki-15021 | Quotient of Cauchy Sequences is Metric Completion | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $d$ be the metric induced by $\struct {R, \norm {\, \cdot \,} }$.
Let $\CC$ be the ring of Cauchy sequences over $R$.
Let $\NN$ be the set of null sequences in $R$.
Let $\CC \,\big / \NN$ be the quotient ring of Cauchy sequences of $\CC$ by the maxi... | By the definition of the metric induced by a norm then:
:a sequence $\sequence {x_n}$ is a Cauchy sequence in $\struct {R, \norm {\, \cdot \,} }$ {{iff}} $\sequence {x_n}$ is a Cauchy sequence in $\struct {R, d}$.
So $\CC$ is the set of Cauchy sequences in $\struct {R, d}$.
Let $\sim$ be the equivalence relation on $\C... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by $\struct {R, \norm {\, \cdot \,} }$.
Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|ring of Cauchy sequences over $R$]... | By the definition of the [[Definition:Metric Induced by Norm on Division Ring|metric induced by a norm]] then:
:a sequence $\sequence {x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in $\struct {R, \norm {\, \cdot \,} }$ {{iff}} $\sequence {x_n}$ is a [[Definition:Cauchy Sequence in M... | Quotient of Cauchy Sequences is Metric Completion | https://proofwiki.org/wiki/Quotient_of_Cauchy_Sequences_is_Metric_Completion | https://proofwiki.org/wiki/Quotient_of_Cauchy_Sequences_is_Metric_Completion | [
"Normed Division Rings",
"Complete Metric Spaces",
"Completion of Normed Division Ring"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence/Normed Division Ring",
"Quotient Ring of Cauchy Sequences is Division Ring",
"Null Sequences form Maximal Left and Right Ideal",
"... | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Cauchy Sequence/Metric Space",
"Equivalence Relation on Cauchy Sequences",
"Definition:Equivalence Class",
"Definition:Equivalence Class",
"De... |
proofwiki-15022 | Completion of Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,}_R }$ be a normed division ring.
Let $\CC$ be the ring of Cauchy sequences over $R$.
Let $\NN$ be the set of null sequences.
Let $Q = \CC / \NN$ where $\CC / \NN$ denotes a quotient ring.
Let $\norm {\, \cdot \,}_Q: Q \to \R_{\ge 0}$ be the norm on the quotient ring $Q$ defined by:... | From Quotient Ring of Cauchy Sequences is Normed Division Ring:
:$\struct {Q, \norm {\, \cdot \,}_Q}$ is a normed division ring.
Let $d_R$ be the metric induced by $\struct {R, \norm {\, \cdot \,}_R }$.
Let $d_Q$ be the metric induced by $\struct {Q, \norm {\, \cdot \,}_Q}$.
From Quotient of Cauchy Sequences is Metric ... | Let $\struct {R, \norm {\, \cdot \,}_R }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|ring of Cauchy sequences over $R$]].
Let $\NN$ be the set of [[Definition:Null Sequence of Normed Division Ring|null sequences]].
Let $Q = \CC / \NN$ where ... | From [[Quotient Ring of Cauchy Sequences is Normed Division Ring]]:
:$\struct {Q, \norm {\, \cdot \,}_Q}$ is a [[Definition:Normed Division Ring|normed division ring]].
Let $d_R$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by $\struct {R, \norm {\, \cdot \,}_R }$.
Let $d_Q$ be the [[D... | Completion of Normed Division Ring | https://proofwiki.org/wiki/Completion_of_Normed_Division_Ring | https://proofwiki.org/wiki/Completion_of_Normed_Division_Ring | [
"Normed Division Rings",
"Complete Metric Spaces",
"Completion of Normed Division Ring"
] | [
"Definition:Normed Division Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Quotient Ring",
"Definition:Norm/Division Ring",
"Definition:Quotient Ring",
"Definition:Completion (Normed Division Ring)"
] | [
"Quotient Ring of Cauchy Sequences is Normed Division Ring",
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Quotient of Cauchy Sequences is Metric Completion",
"Definition:Completion (Metric Space)",
"Defini... |
proofwiki-15023 | Group of Order 27 has Subgroup of Order 3 | Let $G$ be a group whose identity element is $e$.
Let $G$ be of order $27$.
Then $G$ has at least one subgroup of order $3$. | Let $x \in G \setminus \set e$.
From Identity is Only Group Element of Order 1:
:$\order x > 1$
where $\order x$ denotes the order of $x$.
From Lagrange's Theorem, $\order x$ is $3$, $9$ or $27$.
Thus one of the following applies:
{{begin-eqn}}
{{eqn | l = \order x
| r = 3
}}
{{eqn | l = \order {x^3}
| r = ... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $G$ be of [[Definition:Order of Structure|order $27$]].
Then $G$ has at least one [[Definition:Subgroup|subgroup]] of [[Definition:Order of Structure|order $3$]]. | Let $x \in G \setminus \set e$.
From [[Identity is Only Group Element of Order 1]]:
:$\order x > 1$
where $\order x$ denotes the [[Definition:Order of Group Element|order]] of $x$.
From [[Lagrange's Theorem (Group Theory)|Lagrange's Theorem]], $\order x$ is $3$, $9$ or $27$.
Thus one of the following applies:
{{be... | Group of Order 27 has Subgroup of Order 3 | https://proofwiki.org/wiki/Group_of_Order_27_has_Subgroup_of_Order_3 | https://proofwiki.org/wiki/Group_of_Order_27_has_Subgroup_of_Order_3 | [
"Order of Group Elements",
"Groups of Order 27"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Order of Structure",
"Definition:Subgroup",
"Definition:Order of Structure"
] | [
"Identity is Only Group Element of Order 1",
"Definition:Order of Group Element",
"Lagrange's Theorem (Group Theory)",
"Definition:Generated Subgroup",
"Definition:Subgroup",
"Definition:Order of Structure"
] |
proofwiki-15024 | Group does not Necessarily have Subgroup of Order of Divisor of its Order | Let $G$ be a finite group whose order is $n$.
Let $d$ be a divisor of $n$.
Then it is not necessarily the case that $G$ has a subgroup of order $d$. | Proof by Counterexample:
Consider $S_5$, the symmetric group on $5$ letters.
By Order of Symmetric Group, $\order {S_5} = 5! = 120$.
We have that $120 = 8 \times 15$ and so $15$ is a divisor of $120$.
However, Symmetric Group on 5 Letters has no Subgroup of Order 15.
{{qed}} | Let $G$ be a [[Definition:Finite Group|finite group]] whose [[Definition:Order of Structure|order]] is $n$.
Let $d$ be a [[Definition:Divisor of Integer|divisor]] of $n$.
Then it is not necessarily the case that $G$ has a [[Definition:Subgroup|subgroup]] of [[Definition:Order of Structure|order]] $d$. | [[Proof by Counterexample]]:
Consider $S_5$, the [[Definition:Symmetric Group on n Letters|symmetric group on $5$ letters]].
By [[Order of Symmetric Group]], $\order {S_5} = 5! = 120$.
We have that $120 = 8 \times 15$ and so $15$ is a [[Definition:Divisor of Integer|divisor]] of $120$.
However, [[Symmetric Group on... | Group does not Necessarily have Subgroup of Order of Divisor of its Order/Proof 1 | https://proofwiki.org/wiki/Group_does_not_Necessarily_have_Subgroup_of_Order_of_Divisor_of_its_Order | https://proofwiki.org/wiki/Group_does_not_Necessarily_have_Subgroup_of_Order_of_Divisor_of_its_Order/Proof_1 | [
"Group does not Necessarily have Subgroup of Order of Divisor of its Order",
"Subgroups",
"Order of Groups"
] | [
"Definition:Finite Group",
"Definition:Order of Structure",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subgroup",
"Definition:Order of Structure"
] | [
"Proof by Counterexample",
"Definition:Symmetric Group/n Letters",
"Order of Symmetric Group",
"Definition:Divisor (Algebra)/Integer",
"Symmetric Group on 5 Letters has no Subgroup of Order 15"
] |
proofwiki-15025 | Group does not Necessarily have Subgroup of Order of Divisor of its Order | Let $G$ be a finite group whose order is $n$.
Let $d$ be a divisor of $n$.
Then it is not necessarily the case that $G$ has a subgroup of order $d$. | Proof by Counterexample:
Consider the symmetric group $S_4$.
Then the order of the alternating group $A_4$ is $12$.
We list the subgroups of $A_4$:
{{:Alternating Group on 4 Letters/Subgroups}}
Now $6$ divides $12$.
But there is no subgroup of $A_4$ of order $6$.
{{qed}} | Let $G$ be a [[Definition:Finite Group|finite group]] whose [[Definition:Order of Structure|order]] is $n$.
Let $d$ be a [[Definition:Divisor of Integer|divisor]] of $n$.
Then it is not necessarily the case that $G$ has a [[Definition:Subgroup|subgroup]] of [[Definition:Order of Structure|order]] $d$. | [[Proof by Counterexample]]:
Consider the [[Definition:Symmetric Group|symmetric group]] $S_4$.
Then the [[Definition:Order of Structure|order]] of the [[Definition:Alternating Group|alternating group]] $A_4$ is $12$.
We list the [[Alternating Group on 4 Letters/Subgroups|subgroups of $A_4$]]:
{{:Alternating Group ... | Group does not Necessarily have Subgroup of Order of Divisor of its Order/Proof 2 | https://proofwiki.org/wiki/Group_does_not_Necessarily_have_Subgroup_of_Order_of_Divisor_of_its_Order | https://proofwiki.org/wiki/Group_does_not_Necessarily_have_Subgroup_of_Order_of_Divisor_of_its_Order/Proof_2 | [
"Group does not Necessarily have Subgroup of Order of Divisor of its Order",
"Subgroups",
"Order of Groups"
] | [
"Definition:Finite Group",
"Definition:Order of Structure",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subgroup",
"Definition:Order of Structure"
] | [
"Proof by Counterexample",
"Definition:Symmetric Group",
"Definition:Order of Structure",
"Definition:Alternating Group",
"Alternating Group on 4 Letters/Subgroups",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subgroup",
"Definition:Order of Structure"
] |
proofwiki-15026 | Characteristic Function of Normal Distribution | The characteristic function of the normal distribution with mean $\mu$ and variance $\sigma^2$ is given by:
:$\map \phi t = e^{i t \mu - \frac 1 2 t^2 \sigma^2}$ | === {{Lemma|Characteristic Function of Normal Distribution|1}} ===
{{:Characteristic Function of Normal Distribution/Lemma 1}}{{qed|lemma}} | The [[Definition:Characteristic Function of Random Variable|characteristic function]] of the [[Definition:Normal Distribution|normal distribution]] with mean $\mu$ and variance $\sigma^2$ is given by:
:$\map \phi t = e^{i t \mu - \frac 1 2 t^2 \sigma^2}$ | === {{Lemma|Characteristic Function of Normal Distribution|1}} ===
{{:Characteristic Function of Normal Distribution/Lemma 1}}{{qed|lemma}} | Characteristic Function of Normal Distribution | https://proofwiki.org/wiki/Characteristic_Function_of_Normal_Distribution | https://proofwiki.org/wiki/Characteristic_Function_of_Normal_Distribution | [
"Characteristic Function of Normal Distribution",
"Characteristic Functions of Random Variables",
"Normal Distribution"
] | [
"Definition:Characteristic Function of Random Variable",
"Definition:Normal Distribution"
] | [] |
proofwiki-15027 | Product of Subset with Intersection/Equality does not Hold | While it is the case that:
:$X \circ \paren {Y \cap Z} \subseteq \paren {X \circ Y} \cap \paren {X \circ Z}$
it is not necessarily the case that:
:$X \circ \paren {Y \cap Z} = \paren {X \circ Y} \cap \paren {X \circ Z}$ | Proof by Counterexample:
Let $a \in G$ such that $a \ne a^{-1}$.
Let $X = \set {a, a^{-1} }, Y = \set a, Z = \set {a^{-1} }$.
Then:
:$X \circ \paren {Y \cap Z} = X \circ \O = \O$
:$\paren {X \circ Y} \cap \paren {X \circ Z} = \set {a^2, e} \cap \set {e, a^{-2} } \ne \O$
so:
:$X \circ \paren {Y \cap Z} \ne \paren {X \ci... | While it is the case that:
:$X \circ \paren {Y \cap Z} \subseteq \paren {X \circ Y} \cap \paren {X \circ Z}$
it is not necessarily the case that:
:$X \circ \paren {Y \cap Z} = \paren {X \circ Y} \cap \paren {X \circ Z}$ | [[Proof by Counterexample]]:
Let $a \in G$ such that $a \ne a^{-1}$.
Let $X = \set {a, a^{-1} }, Y = \set a, Z = \set {a^{-1} }$.
Then:
:$X \circ \paren {Y \cap Z} = X \circ \O = \O$
:$\paren {X \circ Y} \cap \paren {X \circ Z} = \set {a^2, e} \cap \set {e, a^{-2} } \ne \O$
so:
:$X \circ \paren {Y \cap Z} \ne \pare... | Product of Subset with Intersection/Equality does not Hold | https://proofwiki.org/wiki/Product_of_Subset_with_Intersection/Equality_does_not_Hold | https://proofwiki.org/wiki/Product_of_Subset_with_Intersection/Equality_does_not_Hold | [
"Product of Subset with Intersection"
] | [] | [
"Proof by Counterexample"
] |
proofwiki-15028 | Coset of Subgroup of Subgroup | Let $G$ be a group.
Let $H, K \le G$ be subgroups of $G$.
Let $K \subseteq H$.
Let $x \in G$.
Then either:
:$x K \subseteq H$
or:
:$x K \cap H = \O$
where $x K$ denotes the left coset of $K$ by $x$. | Suppose $x K \cap H \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = x K \cap H
| o = \ne
| r = \O
| c =
}}
{{eqn | ll= \leadsto
| q = \exists y \in G
| l = y
| o = \in
| r = x K \cap H
| c =
}}
{{eqn | ll= \leadsto
| l = y
| o = \in
| r = x K
| c =
}}
... | Let $G$ be a [[Definition:Group|group]].
Let $H, K \le G$ be [[Definition:Subgroup|subgroups]] of $G$.
Let $K \subseteq H$.
Let $x \in G$.
Then either:
:$x K \subseteq H$
or:
:$x K \cap H = \O$
where $x K$ denotes the [[Definition:Left Coset|left coset]] of $K$ by $x$. | Suppose $x K \cap H \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = x K \cap H
| o = \ne
| r = \O
| c =
}}
{{eqn | ll= \leadsto
| q = \exists y \in G
| l = y
| o = \in
| r = x K \cap H
| c =
}}
{{eqn | ll= \leadsto
| l = y
| o = \in
| r = x K
| c =
}... | Coset of Subgroup of Subgroup | https://proofwiki.org/wiki/Coset_of_Subgroup_of_Subgroup | https://proofwiki.org/wiki/Coset_of_Subgroup_of_Subgroup | [
"Cosets"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Coset/Left Coset"
] | [
"Left Cosets are Equal iff Element in Other Left Coset",
"Definition:Group",
"Axiom:Group Axioms",
"Definition:Subset Product"
] |
proofwiki-15029 | Intersection of Left Cosets of Subgroups is Left Coset of Intersection | Let $G$ be a group.
Let $H, K \le G$ be subgroups of $G$.
Let $a, b \in G$.
Let:
:$a H \cap b K \ne \O$
where $a H$ denotes the left coset of $H$ by $a$.
Then $a H \cap b K$ is a left coset of $H \cap K$. | Let $x \in a H \cap b K$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = a H
}}
{{eqn | ll= \leadsto
| l = x H
| r = a H
| c = Left Cosets are Equal iff Element in Other Left Coset
}}
{{end-eqn}}
and similarly:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = b K
}}
{{eqn | ll= \... | Let $G$ be a [[Definition:Group|group]].
Let $H, K \le G$ be [[Definition:Subgroup|subgroups]] of $G$.
Let $a, b \in G$.
Let:
:$a H \cap b K \ne \O$
where $a H$ denotes the [[Definition:Left Coset|left coset]] of $H$ by $a$.
Then $a H \cap b K$ is a [[Definition:Left Coset|left coset]] of $H \cap K$. | Let $x \in a H \cap b K$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = a H
}}
{{eqn | ll= \leadsto
| l = x H
| r = a H
| c = [[Left Cosets are Equal iff Element in Other Left Coset]]
}}
{{end-eqn}}
and similarly:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = b K
}}
{{eqn ... | Intersection of Left Cosets of Subgroups is Left Coset of Intersection | https://proofwiki.org/wiki/Intersection_of_Left_Cosets_of_Subgroups_is_Left_Coset_of_Intersection | https://proofwiki.org/wiki/Intersection_of_Left_Cosets_of_Subgroups_is_Left_Coset_of_Intersection | [
"Cosets"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Coset/Left Coset",
"Definition:Coset/Left Coset"
] | [
"Left Cosets are Equal iff Element in Other Left Coset",
"Left Cosets are Equal iff Element in Other Left Coset",
"Definition:Coset/Left Coset"
] |
proofwiki-15030 | Right Cosets are Equal iff Element in Other Right Coset | Let $H x$ denote the right coset of $H$ by $x$.
Then:
:$H x = H y \iff x \in H y$ | {{begin-eqn}}
{{eqn | l = H x
| r = H y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x y^{-1}
| o = \in
| r = H
| c = Right Cosets are Equal iff Product with Inverse in Subgroup
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \in
| r = H y
| c = Element in Right C... | Let $H x$ denote the [[Definition:Right Coset|right coset]] of $H$ by $x$.
Then:
:$H x = H y \iff x \in H y$ | {{begin-eqn}}
{{eqn | l = H x
| r = H y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x y^{-1}
| o = \in
| r = H
| c = [[Right Cosets are Equal iff Product with Inverse in Subgroup]]
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \in
| r = H y
| c = [[Element in R... | Right Cosets are Equal iff Element in Other Right Coset | https://proofwiki.org/wiki/Right_Cosets_are_Equal_iff_Element_in_Other_Right_Coset | https://proofwiki.org/wiki/Right_Cosets_are_Equal_iff_Element_in_Other_Right_Coset | [
"Cosets"
] | [
"Definition:Coset/Right Coset"
] | [
"Right Cosets are Equal iff Product with Inverse in Subgroup",
"Element in Right Coset iff Product with Inverse in Subgroup"
] |
proofwiki-15031 | Subgroup of Subgroup with Prime Index/Corollary | Let $\struct {G, \circ}$ be a group.
Let $H$ and $K$ be subgroups of $G$.
Let $K \subsetneq H$.
Let:
:$\index G K = p$
where:
:$p$ denotes a prime number
:$\index G K$ denotes the index of $K$ in $G$.
Then:
:$H = G$ | As $K \subsetneq H$ and $K$ is a subgroups of $G$, it follows that $K$ is a proper subgroup of $H$.
That is, $K \ne H$
Hence from Subgroup of Subgroup with Prime Index:
:$H = G$
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $H$ and $K$ be [[Definition:Subgroup|subgroups]] of $G$.
Let $K \subsetneq H$.
Let:
:$\index G K = p$
where:
:$p$ denotes a [[Definition:Prime Number|prime number]]
:$\index G K$ denotes the [[Definition:Index of Subgroup|index]] of $K$ in $G$.
Then:
:... | As $K \subsetneq H$ and $K$ is a [[Definition:Subgroup|subgroups]] of $G$, it follows that $K$ is a [[Definition:Proper Subgroup|proper subgroup]] of $H$.
That is, $K \ne H$
Hence from [[Subgroup of Subgroup with Prime Index]]:
:$H = G$
{{qed}} | Subgroup of Subgroup with Prime Index/Corollary | https://proofwiki.org/wiki/Subgroup_of_Subgroup_with_Prime_Index/Corollary | https://proofwiki.org/wiki/Subgroup_of_Subgroup_with_Prime_Index/Corollary | [
"Index of Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Prime Number",
"Definition:Index of Subgroup"
] | [
"Definition:Subgroup",
"Definition:Proper Subgroup",
"Subgroup of Subgroup with Prime Index"
] |
proofwiki-15032 | Order of Element divides Order of Centralizer | Let $G$ be a finite group.
Let $x \in G$ be an element of $G$.
Let $\map {C_G} x$ denote the centralizer of $x$.
Then:
:$\order x \divides \order {\map {C_G} x}$
where:
:$\order x$ denotes the order of $x$ in $G$
:$\divides$ denotes divisibility
:$\order {\map {C_G} x}$ denotes the order of $\map {C_G} x$. | From Order of Cyclic Group equals Order of Generator:
:$\order x = \order {\gen x}$
where $\gen x$ denotes the subgroup of $G$ generated by $x$.
By definition, $\gen x$ is a cyclic group.
By Cyclic Group is Abelian, all elements of $\gen x$ commute with $x$.
Thus by definition of centralizer:
:$\gen x \subseteq \map {... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $x \in G$ be an [[Definition:Element|element]] of $G$.
Let $\map {C_G} x$ denote the [[Definition:Centralizer of Group Element|centralizer]] of $x$.
Then:
:$\order x \divides \order {\map {C_G} x}$
where:
:$\order x$ denotes the [[Definition:Order of Group ... | From [[Order of Cyclic Group equals Order of Generator]]:
:$\order x = \order {\gen x}$
where $\gen x$ denotes the [[Definition:Generated Subgroup|subgroup of $G$ generated]] by $x$.
By definition, $\gen x$ is a [[Definition:Cyclic Group|cyclic group]].
By [[Cyclic Group is Abelian]], all [[Definition:Element|elemen... | Order of Element divides Order of Centralizer | https://proofwiki.org/wiki/Order_of_Element_divides_Order_of_Centralizer | https://proofwiki.org/wiki/Order_of_Element_divides_Order_of_Centralizer | [
"Centralizers",
"Order of Group Elements"
] | [
"Definition:Finite Group",
"Definition:Element",
"Definition:Centralizer/Group Element",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure"
] | [
"Order of Cyclic Group equals Order of Generator",
"Definition:Generated Subgroup",
"Definition:Cyclic Group",
"Cyclic Group is Abelian",
"Definition:Element",
"Definition:Commutative/Elements",
"Definition:Centralizer/Group Element",
"Centralizer of Group Element is Subgroup",
"Definition:Subgroup"... |
proofwiki-15033 | Left Coset of Stabilizer in Group of Transformations | Let $S$ be a non-empty set.
Let $G$ be a group of permutations of $S$.
Let $t \in G$.
Let $G_t$ be the set defined as:
:$G_t = \set {g \in G: \map g t = t}$
Then each left coset of $G_t$ in $G$ consists of the elements of $G$ that map $t$ to some element of $S$.
{{explain|The source work does not discuss group actions,... | Let $x \in G$.
Consider the left coset $x G_t$.
Let $\map x t = s$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = x G_t
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y^{-1} x
| o = \in
| r = G_t
| c = Element in Left Coset iff Product with Inverse in Subgroup
}}
{{eqn | ll= \... | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]].
Let $G$ be a [[Definition:Permutation Group|group of permutations]] of $S$.
Let $t \in G$.
Let $G_t$ be the [[Definition:Set|set]] defined as:
:$G_t = \set {g \in G: \map g t = t}$
Then each [[Definition:Left Coset|left coset]] of $G_t$ in... | Let $x \in G$.
Consider the [[Definition:Left Coset|left coset]] $x G_t$.
Let $\map x t = s$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = x G_t
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y^{-1} x
| o = \in
| r = G_t
| c = [[Element in Left Coset iff Product with Inve... | Left Coset of Stabilizer in Group of Transformations | https://proofwiki.org/wiki/Left_Coset_of_Stabilizer_in_Group_of_Transformations | https://proofwiki.org/wiki/Left_Coset_of_Stabilizer_in_Group_of_Transformations | [
"Cosets"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Permutation Group",
"Definition:Set",
"Definition:Coset/Left Coset",
"Definition:Element",
"Definition:Element",
"Definition:Stabilizer"
] | [
"Definition:Coset/Left Coset",
"Element in Left Coset iff Product with Inverse in Subgroup"
] |
proofwiki-15034 | Intersection of Coprime Cyclic Subgroups is Trivial | Let $G$ be a group whose identity is $e$.
Let $x, y \in G$ such that:
:$\order x \perp \order y$
where:
:$\order x, \order y$ denotes the orders of $x$ and $y$ in $G$ respectively
:$\perp$ denotes the coprimality relation.
Then:
:$\gen x \cap \gen y = \set e$
where $\gen x, \gen y$ denotes the subgroup of $G$ generated... | From Order of Cyclic Group equals Order of Generator:
:$\order x = \order {\gen x}$
and:
:$\order y = \order {\gen y}$
where $\order {\gen x}, \order {\gen y}$ denote the orders of $\gen x$ and $\gen y$ respectively.
From Intersection of Subgroups is Subgroup:
:$\gen x \cap \gen y$ is a subgroup of both $\gen x$ and $\... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $x, y \in G$ such that:
:$\order x \perp \order y$
where:
:$\order x, \order y$ denotes the [[Definition:Order of Group Element|orders]] of $x$ and $y$ in $G$ respectively
:$\perp$ denotes the [[Definition:Coprime Intege... | From [[Order of Cyclic Group equals Order of Generator]]:
:$\order x = \order {\gen x}$
and:
:$\order y = \order {\gen y}$
where $\order {\gen x}, \order {\gen y}$ denote the [[Definition:Order of Structure|orders]] of $\gen x$ and $\gen y$ respectively.
From [[Intersection of Subgroups is Subgroup]]:
:$\gen x \cap \g... | Intersection of Coprime Cyclic Subgroups is Trivial | https://proofwiki.org/wiki/Intersection_of_Coprime_Cyclic_Subgroups_is_Trivial | https://proofwiki.org/wiki/Intersection_of_Coprime_Cyclic_Subgroups_is_Trivial | [
"Cyclic Groups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Order of Group Element",
"Definition:Coprime/Integers",
"Definition:Generated Subgroup"
] | [
"Order of Cyclic Group equals Order of Generator",
"Definition:Order of Structure",
"Intersection of Subgroups is Subgroup",
"Definition:Subgroup",
"Lagrange's Theorem (Group Theory)",
"Definition:Trivial Subgroup"
] |
proofwiki-15035 | Normed Division Ring is Field iff Completion is Field | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$
Then:
:$R$ is a field {{iff}} $R'$ is a field. | By the definition of a normed division ring completion then:
:$(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R \to R'$.
:$(2): \quad$ $\struct {R', \norm {\, \cdot \,}' }$ is a complete metric space.
:$(3): \quad$ The image $\phi \sqbrk R$ of $\phi$ is a dense subspace in $\struct {R', \norm {... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {R', \norm {\, \cdot \,}' }$ be a [[Definition:Completion (Normed Division Ring)|normed division ring completion]] of $\struct {R, \norm {\, \cdot \,} }$
Then:
:$R$ is a [[Definition:Field (Abstract Alg... | By the definition of a [[Definition:Completion (Normed Division Ring)|normed division ring completion]] then:
:$(1): \quad$ there exists a [[Definition:Distance-Preserving Mapping|distance-preserving]] [[Definition:Ring Monomorphism|ring monomorphism]] $\phi: R \to R'$.
:$(2): \quad$ $\struct {R', \norm {\, \cdot \,}' ... | Normed Division Ring is Field iff Completion is Field | https://proofwiki.org/wiki/Normed_Division_Ring_is_Field_iff_Completion_is_Field | https://proofwiki.org/wiki/Normed_Division_Ring_is_Field_iff_Completion_is_Field | [
"Normed Division Rings",
"Complete Metric Spaces",
"Completion of Normed Division Ring"
] | [
"Definition:Normed Division Ring",
"Definition:Completion (Normed Division Ring)",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Completion (Normed Division Ring)",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Definition:Complete Metric Space",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Everywhere Dense",
"Definition:Topological Subspace",
"Ring Homomorphism Preserves ... |
proofwiki-15036 | Non-Abelian Order 8 Group has Order 4 Element | Let $G$ be a non-abelian group of order $8$.
Then $G$ has at least one element of order $4$. | Let $e \in G$ be the identity of $G$.
Let $g \in G$ be an arbitrary element of $G$ such that $g \ne e$.
From Identity is Only Group Element of Order 1, only $e$ has order $1$.
Thus from Order of Element Divides Order of Finite Group:
:$\order g \in \set {2, 4, 8}$
Suppose $\order g = 8$.
Then $G$ is cyclic.
So by Cycli... | Let $G$ be a non-[[Definition:Abelian Group|abelian]] [[Definition:Group|group]] of [[Definition:Order of Structure|order]] $8$.
Then $G$ has at least one [[Definition:Element|element]] of [[Definition:Order of Group Element|order]] $4$. | Let $e \in G$ be the [[Definition:Identity Element|identity]] of $G$.
Let $g \in G$ be an arbitrary [[Definition:Element|element]] of $G$ such that $g \ne e$.
From [[Identity is Only Group Element of Order 1]], only $e$ has [[Definition:Order of Group Element|order]] $1$.
Thus from [[Order of Element Divides Order o... | Non-Abelian Order 8 Group has Order 4 Element | https://proofwiki.org/wiki/Non-Abelian_Order_8_Group_has_Order_4_Element | https://proofwiki.org/wiki/Non-Abelian_Order_8_Group_has_Order_4_Element | [
"Groups of Order 8"
] | [
"Definition:Abelian Group",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Element",
"Definition:Order of Group Element"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Identity is Only Group Element of Order 1",
"Definition:Order of Group Element",
"Order of Element Divides Order of Finite Group",
"Definition:Cyclic Group",
"Cyclic Group is Abelian",
"Definition:Abelian Group",
"De... |
proofwiki-15037 | Group of Prime Order p has p-1 Elements of Order p | Let $p$ be a prime number.
Let $G$ be a group with identity $e$ whose order is $p$.
Then $G$ has $p - 1$ elements of order $p$. | Let $\order g$ denote the order of an element $g$ of $G$.
From Order of Element Divides Order of Finite Group:
:$\order g \divides p$
where $\divides$ denotes divisibility.
By definition of prime number, the only divisors of $p$ are $1$ and $p$.
From Identity is Only Group Element of Order 1, only $e$ has order $1$ in ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Group|group]] with [[Definition:Identity Element|identity]] $e$ whose [[Definition:Order of Structure|order]] is $p$.
Then $G$ has $p - 1$ [[Definition:Element|elements]] of [[Definition:Order of Group Element|order]] $p$. | Let $\order g$ denote the [[Definition:Order of Group Element|order]] of an [[Definition:Element|element]] $g$ of $G$.
From [[Order of Element Divides Order of Finite Group]]:
:$\order g \divides p$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
By definition of [[Definition:Prime Number|pri... | Group of Prime Order p has p-1 Elements of Order p | https://proofwiki.org/wiki/Group_of_Prime_Order_p_has_p-1_Elements_of_Order_p | https://proofwiki.org/wiki/Group_of_Prime_Order_p_has_p-1_Elements_of_Order_p | [
"Prime Groups",
"Order of Group Elements"
] | [
"Definition:Prime Number",
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Order of Structure",
"Definition:Element",
"Definition:Order of Group Element"
] | [
"Definition:Order of Group Element",
"Definition:Element",
"Order of Element Divides Order of Finite Group",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Identity is Only Group Element of Order 1",
"Definition:Order of Group Element",
"D... |
proofwiki-15038 | Number of Order p Elements in Group with m Order p Subgroups | Let $G$ be a group whose identity is $e$.
Let $G$ have $m$ subgroups of order $p$.
The total number of elements of $G$ of order $p$ is $m \paren {p - 1}$. | Let $H \le G$ be a subgroup of $G$ of order $p$.
From Prime Group is Cyclic, $H$ is a cyclic group.
From Group of Prime Order p has p-1 Elements of Order p, $H$ has $p - 1$ elements of order $p$.
From Intersection of Subgroups of Prime Order, each of the $m$ sets of $p - 1$ non-identity elements of the $m$ subgroups of... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $G$ have $m$ [[Definition:Subgroup|subgroups]] of [[Definition:Order of Structure|order]] $p$.
The total number of [[Definition:Element|elements]] of $G$ of [[Definition:Order of Group Element|order]] $p$ is $m \paren ... | Let $H \le G$ be a [[Definition:Subgroup|subgroup]] of $G$ of [[Definition:Order of Structure|order]] $p$.
From [[Prime Group is Cyclic]], $H$ is a [[Definition:Cyclic Group|cyclic group]].
From [[Group of Prime Order p has p-1 Elements of Order p]], $H$ has $p - 1$ [[Definition:Element|elements]] of [[Definition:Ord... | Number of Order p Elements in Group with m Order p Subgroups | https://proofwiki.org/wiki/Number_of_Order_p_Elements_in_Group_with_m_Order_p_Subgroups | https://proofwiki.org/wiki/Number_of_Order_p_Elements_in_Group_with_m_Order_p_Subgroups | [
"Order of Group Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Element",
"Definition:Order of Group Element"
] | [
"Definition:Subgroup",
"Definition:Order of Structure",
"Prime Group is Cyclic",
"Definition:Cyclic Group",
"Group of Prime Order p has p-1 Elements of Order p",
"Definition:Element",
"Definition:Order of Group Element",
"Intersection of Subgroups of Prime Order",
"Definition:Set",
"Definition:Ide... |
proofwiki-15039 | Non-Cyclic Group of Order p^2 has p+3 Subgroups | Let $p$ be a prime number.
Let $G$ be a non-cyclic group whose order is $p^2$.
Then $G$ has exactly $p + 3$ subgroups. | By Order of Element Divides Order of Finite Group, all elements of $G$ have order in $\set {1, p, p^2}$.
But as $G$ is non-cyclic, it can have no element of order $p^2$.
By Identity is Only Group Element of Order 1, $G$ has $p^2 - 1$ elements of order $p$.
Let $m$ denote the number of subgroups of $G$ of order $p$.
Fro... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a non-[[Definition:Cyclic Group|cyclic group]] whose [[Definition:Order of Structure|order]] is $p^2$.
Then $G$ has exactly $p + 3$ [[Definition:Subgroup|subgroups]]. | By [[Order of Element Divides Order of Finite Group]], all [[Definition:Element|elements]] of $G$ have [[Definition:Order of Group Element|order]] in $\set {1, p, p^2}$.
But as $G$ is non-[[Definition:Cyclic Group|cyclic]], it can have no [[Definition:Element|element]] of [[Definition:Order of Group Element|order]] $p... | Non-Cyclic Group of Order p^2 has p+3 Subgroups | https://proofwiki.org/wiki/Non-Cyclic_Group_of_Order_p^2_has_p+3_Subgroups | https://proofwiki.org/wiki/Non-Cyclic_Group_of_Order_p^2_has_p+3_Subgroups | [
"Examples of Order of Group Elements"
] | [
"Definition:Prime Number",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Subgroup"
] | [
"Order of Element Divides Order of Finite Group",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Cyclic Group",
"Definition:Element",
"Definition:Order of Group Element",
"Identity is Only Group Element of Order 1",
"Definition:Element",
"Definition:Order of Group Element",
... |
proofwiki-15040 | Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2 | Let $p$ be a prime number.
Let $G$ be a finite abelian group whose identity is $e$.
Let $G$ have at least $p$ elements of order $p$.
Then:
: $p^2 \divides \order G$
where:
:$\divides$ denotes divisibility
:$\order G$ denotes the order of $G$. | Let $x \in G$ be of order $p$.
Consider $\gen x$, the subgroup generated by $x$.
By Group of Prime Order p has p-1 Elements of Order p, the elements of $\gen x$ are all of order $p$ except $e$.
Thus, by hypothesis, there must exist another $y \in G$ of order $p$.
Consider the subset of $G$:
:$S := \set {x^i y^j: 0 \le ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Finite Group|finite]] [[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $G$ have at least $p$ [[Definition:Element|elements]] of [[Definition:Order of Group Element|order]] $p$.
Then:
... | Let $x \in G$ be of [[Definition:Order of Group Element|order]] $p$.
Consider $\gen x$, the [[Definition:Generated Subgroup|subgroup generated]] by $x$.
By [[Group of Prime Order p has p-1 Elements of Order p]], the [[Definition:Element|elements]] of $\gen x$ are all of [[Definition:Order of Group Element|order]] $p$... | Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2 | https://proofwiki.org/wiki/Order_of_Finite_Abelian_Group_with_p+_Order_p_Elements_is_Divisible_by_p^2 | https://proofwiki.org/wiki/Order_of_Finite_Abelian_Group_with_p+_Order_p_Elements_is_Divisible_by_p^2 | [
"Order of Group Elements",
"Examples of Order of Group Elements",
"Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2",
"Abelian Groups",
"Finite Groups"
] | [
"Definition:Prime Number",
"Definition:Finite Group",
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure"
] | [
"Definition:Order of Group Element",
"Definition:Generated Subgroup",
"Group of Prime Order p has p-1 Elements of Order p",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:By Hypothesis",
"Definition:Order of Group Element",
"Definition:Subset",
"Finite Subgroup Test",
"Defi... |
proofwiki-15041 | Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2 | Let $p$ be a prime number.
Let $G$ be a finite abelian group whose identity is $e$.
Let $G$ have at least $p$ elements of order $p$.
Then:
: $p^2 \divides \order G$
where:
:$\divides$ denotes divisibility
:$\order G$ denotes the order of $G$. | By hypothesis there are elements $x, y$ of order $3$ in $G$ such that $x, y, x^2$ are all different.
Consider the subset of $G$:
:$S := \set {x^i y^j: 0 \le i, j \le 2}$
By the Finite Subgroup Test, $S$ is a subgroup of $G$ which has $9$ elements.
By Lagrange's theorem:
:$\order S \divides \order G$
But $\order S = 9$ ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Finite Group|finite]] [[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $G$ have at least $p$ [[Definition:Element|elements]] of [[Definition:Order of Group Element|order]] $p$.
Then:
... | [[Definition:By Hypothesis|By hypothesis]] there are [[Definition:Element|elements]] $x, y$ of [[Definition:Order of Group Element|order]] $3$ in $G$ such that $x, y, x^2$ are all different.
Consider the [[Definition:Subset|subset]] of $G$:
:$S := \set {x^i y^j: 0 \le i, j \le 2}$
By the [[Finite Subgroup Test]], $S$... | Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2/Examples/Order 3/Proof 1 | https://proofwiki.org/wiki/Order_of_Finite_Abelian_Group_with_p+_Order_p_Elements_is_Divisible_by_p^2 | https://proofwiki.org/wiki/Order_of_Finite_Abelian_Group_with_p+_Order_p_Elements_is_Divisible_by_p^2/Examples/Order_3/Proof_1 | [
"Order of Group Elements",
"Examples of Order of Group Elements",
"Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2",
"Abelian Groups",
"Finite Groups"
] | [
"Definition:Prime Number",
"Definition:Finite Group",
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure"
] | [
"Definition:By Hypothesis",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Subset",
"Finite Subgroup Test",
"Definition:Subgroup",
"Definition:Element",
"Lagrange's Theorem (Group Theory)"
] |
proofwiki-15042 | Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2 | Let $p$ be a prime number.
Let $G$ be a finite abelian group whose identity is $e$.
Let $G$ have at least $p$ elements of order $p$.
Then:
: $p^2 \divides \order G$
where:
:$\divides$ denotes divisibility
:$\order G$ denotes the order of $G$. | An example of Order of Finite Abelian Group with $p+$ Order $p$ Elements is Divisible by $p^2$, setting $p = 3$.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Finite Group|finite]] [[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $G$ have at least $p$ [[Definition:Element|elements]] of [[Definition:Order of Group Element|order]] $p$.
Then:
... | An example of [[Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2|Order of Finite Abelian Group with $p+$ Order $p$ Elements is Divisible by $p^2$]], setting $p = 3$.
{{qed}} | Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2/Examples/Order 3/Proof 2 | https://proofwiki.org/wiki/Order_of_Finite_Abelian_Group_with_p+_Order_p_Elements_is_Divisible_by_p^2 | https://proofwiki.org/wiki/Order_of_Finite_Abelian_Group_with_p+_Order_p_Elements_is_Divisible_by_p^2/Examples/Order_3/Proof_2 | [
"Order of Group Elements",
"Examples of Order of Group Elements",
"Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2",
"Abelian Groups",
"Finite Groups"
] | [
"Definition:Prime Number",
"Definition:Finite Group",
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure"
] | [
"Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2"
] |
proofwiki-15043 | Abelian Group of Semiprime Order is Cyclic | Let $p$ and $q$ be distinct prime numbers.
Let $G$ be an abelian group such that:
:$\order G = p q$
where $\order G$ denotes the order of $G$.
Then $G$ is cyclic. | By Order of Element Divides Order of Finite Group, the order of elements of $G$ are all in $\set {1, p, q, p q}$.
We have that Identity is Only Group Element of Order 1.
Suppose $G$ were to contain more than $p - 1$ elements of order $p$.
Then by Order of Finite Abelian Group with $p+$ Order $p$ Elements is Divisible b... | Let $p$ and $q$ be [[Definition:Distinct|distinct]] [[Definition:Prime Number|prime numbers]].
Let $G$ be an [[Definition:Abelian Group|abelian group]] such that:
:$\order G = p q$
where $\order G$ denotes the [[Definition:Order of Structure|order]] of $G$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | By [[Order of Element Divides Order of Finite Group]], the [[Definition:Order of Group Element|order]] of [[Definition:Element|elements]] of $G$ are all in $\set {1, p, q, p q}$.
We have that [[Identity is Only Group Element of Order 1]].
Suppose $G$ were to contain more than $p - 1$ [[Definition:Element|elements]] ... | Abelian Group of Semiprime Order is Cyclic | https://proofwiki.org/wiki/Abelian_Group_of_Semiprime_Order_is_Cyclic | https://proofwiki.org/wiki/Abelian_Group_of_Semiprime_Order_is_Cyclic | [
"Abelian Groups",
"Finite Groups",
"Cyclic Groups"
] | [
"Definition:Distinct",
"Definition:Prime Number",
"Definition:Abelian Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Order of Element Divides Order of Finite Group",
"Definition:Order of Group Element",
"Definition:Element",
"Identity is Only Group Element of Order 1",
"Definition:Element",
"Definition:Order of Group Element",
"Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2",
"Definition... |
proofwiki-15044 | General Morphism Property for Groups | Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: G \to H$ be a group homomorphism.
Then:
:$\forall g_k \in H: \map \phi {g_1 \circ g_2 \circ \cdots \circ g_n} = \map \phi {g_1} * \map \phi {g_2} * \cdots * \map \phi {g_n}$ | A group is a semigroup.
The result then follows from the General Morphism Property for Semigroups.
{{Qed}} | Let $\struct {G, \circ}$ and $\struct {H, *}$ be [[Definition:Group|groups]].
Let $\phi: G \to H$ be a [[Definition:Group Homomorphism|group homomorphism]].
Then:
:$\forall g_k \in H: \map \phi {g_1 \circ g_2 \circ \cdots \circ g_n} = \map \phi {g_1} * \map \phi {g_2} * \cdots * \map \phi {g_n}$ | A [[Definition:Group|group]] is a [[Definition:Semigroup|semigroup]].
The result then follows from the [[General Morphism Property for Semigroups]].
{{Qed}} | General Morphism Property for Groups | https://proofwiki.org/wiki/General_Morphism_Property_for_Groups | https://proofwiki.org/wiki/General_Morphism_Property_for_Groups | [
"Morphism Property",
"Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Group Homomorphism"
] | [
"Definition:Group",
"Definition:Semigroup",
"General Morphism Property for Semigroups"
] |
proofwiki-15045 | General Linear Group to Determinant is Homomorphism/Corollary | The kernel of the $\det$ mapping is the special linear group $\SL {n, \R}$. | From General Linear Group to Determinant is Homomorphism:
:$\det$ is a group homomorphism.
The special linear group $\SL {n, \R}$ is the subset of $\GL {n, \R}$ such that:
:$\forall \mathbf A \in \SL {n, \R}: \map \det {\mathbf A} = 1$
From Real Multiplication Identity is One:
: $1$ is the identity of the multiplicativ... | The [[Definition:Kernel of Group Homomorphism|kernel]] of the $\det$ [[Definition:Mapping|mapping]] is the [[Definition:Special Linear Group|special linear group]] $\SL {n, \R}$. | From [[General Linear Group to Determinant is Homomorphism]]:
:$\det$ is a [[Definition:Group Homomorphism|group homomorphism]].
The [[Definition:Special Linear Group|special linear group]] $\SL {n, \R}$ is the [[Definition:Subset|subset]] of $\GL {n, \R}$ such that:
:$\forall \mathbf A \in \SL {n, \R}: \map \det {\m... | General Linear Group to Determinant is Homomorphism/Corollary | https://proofwiki.org/wiki/General_Linear_Group_to_Determinant_is_Homomorphism/Corollary | https://proofwiki.org/wiki/General_Linear_Group_to_Determinant_is_Homomorphism/Corollary | [
"Examples of Group Homomorphisms",
"General Linear Group",
"Special Linear Group",
"Determinants"
] | [
"Definition:Kernel of Group Homomorphism",
"Definition:Mapping",
"Definition:Special Linear Group"
] | [
"General Linear Group to Determinant is Homomorphism",
"Definition:Group Homomorphism",
"Definition:Special Linear Group",
"Definition:Subset",
"Real Multiplication Identity is One",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multiplicative Group of Real Numbers",
"Defini... |
proofwiki-15046 | Quotient Group of General Linear Group by Special Linear Group | Let $\GL {n, \R}$ denote the general linear group of degree $n$ over $\R$.
Let $\SL {n, \R}$ denote the special linear group of degree $n$ over $\R$.
Then the quotient group $\GL {n, \R} / \SL {n, \R}$ is the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$. | Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the group homomorphism:
:$\mathbf A \mapsto \map \det {\mathbf A}$
where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$.
This is demonstrated to be a homomorphism in General Linear Group to Determinant is Homomorphism
From {{Corollary|General Linear... | Let $\GL {n, \R}$ denote the [[Definition:General Linear Group|general linear group]] of degree $n$ over $\R$.
Let $\SL {n, \R}$ denote the [[Definition:Special Linear Group|special linear group]] of degree $n$ over $\R$.
Then the [[Definition:Quotient Group|quotient group]] $\GL {n, \R} / \SL {n, \R}$ is the [[Defi... | Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the [[Definition:Group Homomorphism|group homomorphism]]:
:$\mathbf A \mapsto \map \det {\mathbf A}$
where $\map \det {\mathbf A}$ is the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
This is demonstrated to be a [[Definition:Group Homomorp... | Quotient Group of General Linear Group by Special Linear Group | https://proofwiki.org/wiki/Quotient_Group_of_General_Linear_Group_by_Special_Linear_Group | https://proofwiki.org/wiki/Quotient_Group_of_General_Linear_Group_by_Special_Linear_Group | [
"General Linear Group",
"Special Linear Group",
"Examples of Quotient Groups"
] | [
"Definition:General Linear Group",
"Definition:Special Linear Group",
"Definition:Quotient Group",
"Definition:Multiplicative Group of Real Numbers"
] | [
"Definition:Group Homomorphism",
"Definition:Determinant/Matrix",
"Definition:Group Homomorphism",
"General Linear Group to Determinant is Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Kernel is Normal Subgroup of Domain",
"Definition:Normal Subgroup",
"First Isomorphism Theorem/Groups",
... |
proofwiki-15047 | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 1 | Let $\psi' = \phi_2 \circ \phi_1^{-1}:\phi_1 \paren R \to \phi_2 \paren R$ be the composition of $\phi_1^{-1}$ with $\phi_2$.
Then $\psi': \struct {\map {\phi_1} R, \norm {\, \cdot \,}_1 } \to \struct {\map {\phi_2} R, \norm {\, \cdot \,}_2 }$ is an isometric ring isomorphism. | By Monomorphism Image is Isomorphic to Domain, $\phi_1:R \to \map {\phi_1} R$ and $\phi_2:R \to \map {\phi_2} R$ are ring isomorphisms.
By Distance-Preserving Image Isometric to Domain for Metric Spaces, $\phi_1:R \to \map {\phi_1} R$ and $\phi_2:R \to \map {\phi_2} R$ are isometries.
By Inverse of Algebraic Structure ... | Let $\psi' = \phi_2 \circ \phi_1^{-1}:\phi_1 \paren R \to \phi_2 \paren R$ be the composition of $\phi_1^{-1}$ with $\phi_2$.
Then $\psi': \struct {\map {\phi_1} R, \norm {\, \cdot \,}_1 } \to \struct {\map {\phi_2} R, \norm {\, \cdot \,}_2 }$ is an [[Definition:Isometry (Metric Spaces)|isometric]] [[Definition:Ring ... | By [[Monomorphism Image is Isomorphic to Domain]], $\phi_1:R \to \map {\phi_1} R$ and $\phi_2:R \to \map {\phi_2} R$ are [[Definition:Ring Isomorphism|ring isomorphisms]].
By [[Distance-Preserving Image Isometric to Domain for Metric Spaces]], $\phi_1:R \to \map {\phi_1} R$ and $\phi_2:R \to \map {\phi_2} R$ are [[Def... | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 1 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_1 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_1 | [
"Normed Division Ring Completions are Isometric and Isomorphic"
] | [
"Definition:Isometry (Metric Spaces)",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism"
] | [
"Monomorphism Image is Isomorphic to Domain",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Distance-Preserving Image Isometric to Domain for Metric Spaces",
"Definition:Isometry (Metric Spaces)",
"Inverse of Algebraic Structure Isomorphism is Isomorphism",
"Definition:Isomorphism (Abstra... |
proofwiki-15048 | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 2 | Let $\psi: S_1 \to S_2$ be defined by:
:$\forall x \in S_1: \map \psi x = \ds \lim_{n \mathop \to \infty} \map {\psi'} {x_n}$
where $\ds x = \lim_{n \mathop \to \infty} x_n$ for some sequence $\sequence {x_n} \subseteq R_1$
Then $\psi$ is a well-defined mapping. | Let $x \in S_1$.
By the definition of dense subset:
:$\map \cl {R_1} = S_1$
By Closure of Subset of Metric Space by Convergent Sequence, there exists a sequence $\sequence {x_n} \subseteq R_1 $ that converges to $x$, that is:
:$\ds \lim_{n \mathop \to \infty} x_n = x$
By Isometric Image of Cauchy Sequence is Cauchy Seq... | Let $\psi: S_1 \to S_2$ be defined by:
:$\forall x \in S_1: \map \psi x = \ds \lim_{n \mathop \to \infty} \map {\psi'} {x_n}$
where $\ds x = \lim_{n \mathop \to \infty} x_n$ for some [[Definition:Sequence|sequence]] $\sequence {x_n} \subseteq R_1$
Then $\psi$ is a [[Definition:Well-Defined Mapping|well-defined mapping... | Let $x \in S_1$.
By the definition of [[Definition:Dense|dense subset]]:
:$\map \cl {R_1} = S_1$
By [[Closure of Subset of Metric Space by Convergent Sequence]], there exists a [[Definition:Sequence|sequence]] $\sequence {x_n} \subseteq R_1 $ that [[Definition:Convergent Sequence in Normed Division Ring|converges]] t... | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 2 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_2 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_2 | [
"Normed Division Ring Completions are Isometric and Isomorphic"
] | [
"Definition:Sequence",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Dense",
"Closure of Subset of Metric Space by Convergent Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Isometric Image of Cauchy Sequence is Cauchy Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Complete Normed Division R... |
proofwiki-15049 | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 3 | :$\psi$ is a surjective mapping. | Let $y \in S_2$.
By the definition of dense subset:
:$\map \cl {R_2} = S_2$
By Closure of Subset of Metric Space by Convergent Sequence:
:there exists a sequence $\sequence {y_n} \subseteq R_2 $ that converges to $y$, that is, $\ds \lim_{n \mathop \to \infty} y_n = y$
By Isometric Image of Cauchy Sequence is Cauchy Seq... | :$\psi$ is a [[Definition:Surjection|surjective mapping]]. | Let $y \in S_2$.
By the definition of [[Definition:Dense|dense subset]]:
:$\map \cl {R_2} = S_2$
By [[Closure of Subset of Metric Space by Convergent Sequence]]:
:there exists a [[Definition:Sequence|sequence]] $\sequence {y_n} \subseteq R_2 $ that [[Definition:Convergent Sequence in Normed Division Ring|converges]] ... | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 3 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_3 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_3 | [
"Normed Division Ring Completions are Isometric and Isomorphic"
] | [
"Definition:Surjection"
] | [
"Definition:Dense",
"Closure of Subset of Metric Space by Convergent Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Isometric Image of Cauchy Sequence is Cauchy Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Complete Normed Division R... |
proofwiki-15050 | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 4 | :$\psi$ is an isometry. | Let $x, y \in S_1$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $R_1$ such that $\ds \lim_{n \mathop \to \infty} x_n = x, \lim_{n \mathop \to \infty} y_n = y$.
Then:
{{begin-eqn}}
{{eqn | l = x - y
| r = \lim_{n \mathop \to \infty} x_n - y_n
| c = Difference Rule for Sequences in Normed Divi... | :$\psi$ is an [[Definition:Isometry (Metric Spaces)|isometry]]. | Let $x, y \in S_1$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Sequence|sequences]] in $R_1$ such that $\ds \lim_{n \mathop \to \infty} x_n = x, \lim_{n \mathop \to \infty} y_n = y$.
Then:
{{begin-eqn}}
{{eqn | l = x - y
| r = \lim_{n \mathop \to \infty} x_n - y_n
| c = [[Difference Rule... | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 4 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_4 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_4 | [
"Normed Division Ring Completions are Isometric and Isomorphic"
] | [
"Definition:Isometry (Metric Spaces)"
] | [
"Definition:Sequence",
"Combination Theorem for Sequences/Normed Division Ring/Difference Rule",
"Modulus of Limit/Normed Division Ring",
"Combination Theorem for Sequences/Normed Division Ring/Difference Rule",
"Modulus of Limit/Normed Division Ring",
"Convergent Sequence in Metric Space has Unique Limit... |
proofwiki-15051 | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 5 | :$\psi$ is a ring isomorphism. | By {{Lemma|Normed Division Ring Completions are Isometric and Isomorphic|4}}, $\psi$ is an isometry.
By the definition of an isometry, $\psi$ is a bijection.
By the definition of a ring isomorphism, all that remains is to show that $\psi$ is a ring homomorphism.
That is:
:$(1): \quad \forall x, y \in S_1: \map \psi {x ... | :$\psi$ is a [[Definition:Ring Isomorphism|ring isomorphism]]. | By {{Lemma|Normed Division Ring Completions are Isometric and Isomorphic|4}}, $\psi$ is an [[Definition:Isometry (Metric Spaces)|isometry]].
By the definition of an [[Definition:Isometry (Metric Spaces)|isometry]], $\psi$ is a [[Definition:Bijective|bijection]].
By the definition of a [[Definition:Ring Isomorphism|ri... | Normed Division Ring Completions are Isometric and Isomorphic/Lemma 5 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_5 | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic/Lemma_5 | [
"Normed Division Ring Completions are Isometric and Isomorphic"
] | [
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism"
] | [
"Definition:Isometry (Metric Spaces)",
"Definition:Isometry (Metric Spaces)",
"Definition:Bijection",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Ring Homomorphism",
"Definition:Sequence",
"Definition:Sequence",
"Combination Theorem for Sequences/Normed Division Ring/Sum ... |
proofwiki-15052 | Normed Division Ring Completions are Isometric and Isomorphic | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {S_1, \norm {\, \cdot \,}_1 }$ and $\struct {S_2, \norm {\, \cdot \,}_2 }$ be normed division ring completions of $\struct {R, \norm {\, \cdot \,} }$
Then there exists an isometric isomorphism:
:$\psi: \struct {S_1, \norm {\, \cdot \,}_1 } ... | By the definition of a normed division ring completion then:
:there exists a distance-preserving ring monomorphisms $\phi_1: R \to S_1$
:$R_1 = \map {\phi_1} R$ is a dense subring of $S_1$
:$S_1$ is a complete metric space
:there exists a distance-preserving ring monomorphisms $\phi_2: R \to S_2$
:$R_2 = \map {\phi_2} ... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {S_1, \norm {\, \cdot \,}_1 }$ and $\struct {S_2, \norm {\, \cdot \,}_2 }$ be [[Definition:Completion (Normed Division Ring)|normed division ring completions]] of $\struct {R, \norm {\, \cdot \,} }$
Th... | By the definition of a [[Definition:Completion (Normed Division Ring)|normed division ring completion]] then:
:there exists a [[Definition:Distance-Preserving Mapping|distance-preserving]] [[Definition:Ring Monomorphism|ring monomorphisms]] $\phi_1: R \to S_1$
:$R_1 = \map {\phi_1} R$ is a [[Definition:Dense|dense]] [[... | Normed Division Ring Completions are Isometric and Isomorphic | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic | https://proofwiki.org/wiki/Normed_Division_Ring_Completions_are_Isometric_and_Isomorphic | [
"Normed Division Ring Completions are Isometric and Isomorphic",
"Completion of Normed Division Ring",
"Normed Division Rings",
"Complete Metric Spaces"
] | [
"Definition:Normed Division Ring",
"Definition:Completion (Normed Division Ring)",
"Definition:Isometric Isomorphism/Normed Division Ring"
] | [
"Definition:Completion (Normed Division Ring)",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Definition:Dense",
"Definition:Subring",
"Definition:Complete Metric Space",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Definition:Dense",
"Def... |
proofwiki-15053 | Distance-Preserving Image Isometric to Domain for Metric Spaces | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $\phi: M_1 \to M_2$ be a distance-preserving mapping.
Then:
:$\phi: M_1 \to \Img \phi$
is an isometry. | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $\phi$ be a distance-preserving mapping from $M_1$ to $M_2$.
Let $A = \Img \phi$ be the image of $\phi$.
By Subspace of Metric Space is Metric Space, $\struct {A, d_2}$ is a metric space.
As $\phi$ is a distance-preserving mapping, by D... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $\phi: M_1 \to M_2$ be a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]].
Then:
:$\phi: M_1 \to \Img \phi$
is an [[Definition:Isometry (Metric Spaces)|isometry]]. | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $\phi$ be a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]] from $M_1$ to $M_2$.
Let $A = \Img \phi$ be the [[Definition:Image of Mapping|image]] of $\phi$.
By [[Subspace of Metric S... | Distance-Preserving Image Isometric to Domain for Metric Spaces | https://proofwiki.org/wiki/Distance-Preserving_Image_Isometric_to_Domain_for_Metric_Spaces | https://proofwiki.org/wiki/Distance-Preserving_Image_Isometric_to_Domain_for_Metric_Spaces | [
"Isometries (Metric Spaces)",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Distance-Preserving Mapping",
"Definition:Isometry (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Distance-Preserving Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Subspace of Metric Space is Metric Space",
"Definition:Metric Space",
"Definition:Distance-Preserving Mapping",
"Distance-Preserving Mapping is Injection of Metric Spaces",
"Definitio... |
proofwiki-15054 | Distance-Preserving Mapping is Injection of Metric Spaces | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $\phi: M_1 \to M_2$ be a distance-preserving mapping.
Then $\phi$ is an injection. | Let $a, b \in A_1$ and suppose that $\map \phi a = \map \phi b$.
Then by the definition of a metric space:
:$\map {d_2} {\map \phi a, \map \phi b} = 0$
By the definition of a distance-preserving mapping then:
:$\map {d_1} {a, b} = 0$
Thus by the definition of a metric space:
:$a = b$
Hence $\phi$ is injective.
{{qed}}
... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $\phi: M_1 \to M_2$ be a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]].
Then $\phi$ is an [[Definition:Injective|injection]]. | Let $a, b \in A_1$ and suppose that $\map \phi a = \map \phi b$.
Then by the definition of a [[Definition:Metric Space|metric space]]:
:$\map {d_2} {\map \phi a, \map \phi b} = 0$
By the definition of a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]] then:
:$\map {d_1} {a, b} = 0$
Thus by th... | Distance-Preserving Mapping is Injection of Metric Spaces | https://proofwiki.org/wiki/Distance-Preserving_Mapping_is_Injection_of_Metric_Spaces | https://proofwiki.org/wiki/Distance-Preserving_Mapping_is_Injection_of_Metric_Spaces | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Distance-Preserving Mapping",
"Definition:Injective"
] | [
"Definition:Metric Space",
"Definition:Distance-Preserving Mapping",
"Definition:Metric Space",
"Definition:Injection",
"Category:Metric Spaces"
] |
proofwiki-15055 | Odd Power Function is Surjective | Let $n \in \Z_{\ge 0}$ be an odd positive integer.
Let $f_n: \R \to \R$ be the real function defined as:
:$\map {f_n} x = x^n$
Then $f_n$ is a surjection. | From Existence of Positive Root of Positive Real Number we have that:
:$\forall x \in \R_{\ge 0}: \exists y \in \R: y^n = x$
From Power of Ring Negative:
:$\paren {-x}^n = -\paren {x^n}$
and so:
:$\forall x \in \R_{\le 0}: \exists y \in \R: y^n = x$
Thus:
:$\forall x \in \R: \exists y \in \R: y^n = x$
and so $f_n$ is a... | Let $n \in \Z_{\ge 0}$ be an [[Definition:Odd Integer|odd]] [[Definition:Strictly Positive Integer|positive integer]].
Let $f_n: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map {f_n} x = x^n$
Then $f_n$ is a [[Definition:Surjection|surjection]]. | From [[Existence of Positive Root of Positive Real Number]] we have that:
:$\forall x \in \R_{\ge 0}: \exists y \in \R: y^n = x$
From [[Power of Ring Negative]]:
:$\paren {-x}^n = -\paren {x^n}$
and so:
:$\forall x \in \R_{\le 0}: \exists y \in \R: y^n = x$
Thus:
:$\forall x \in \R: \exists y \in \R: y^n = x$
and so ... | Odd Power Function is Surjective | https://proofwiki.org/wiki/Odd_Power_Function_is_Surjective | https://proofwiki.org/wiki/Odd_Power_Function_is_Surjective | [
"Powers",
"Surjections",
"Real Functions"
] | [
"Definition:Odd Integer",
"Definition:Strictly Positive/Integer",
"Definition:Real Function",
"Definition:Surjection"
] | [
"Existence of Positive Root of Positive Real Number",
"Power of Ring Negative",
"Definition:Surjection",
"Category:Powers",
"Category:Surjections",
"Category:Real Functions"
] |
proofwiki-15056 | Residue at Multiple Pole | Let $f: \C \to \C$ be a function meromorphic on some region, $D$, containing $a$.
Let $f$ have a single pole in $D$, of order $N$, at $a$.
Then the residue of $f$ at $a$ is given by:
:$\ds \Res f a = \frac 1 {\paren {N - 1}!} \lim_{z \mathop \to a} \frac {\d^{N - 1} } {\d z^{N - 1} } \paren {\paren {z - a}^N \map f z}... | By Existence of Laurent Series, there exists a Laurent series:
:$\ds \map f z = \sum_{n \mathop = -\infty}^\infty c_n \paren {z - a}^n$
convergent on $D \setminus \set a$.
As $f$ has a pole of order $N$ at $a$, we have $c_n = 0$ for $n < -N$.
So:
:$\ds \paren {z - a}^N \map f z = \sum_{n \mathop = -N}^\infty c_n \... | Let $f: \C \to \C$ be a [[Definition:Complex Function|function]] [[Definition:Meromorphic Function|meromorphic]] on some [[Definition:Region (Complex Analysis)|region]], $D$, containing $a$.
Let $f$ have a single [[Definition:Pole|pole]] in $D$, of order $N$, at $a$.
Then the [[Definition:Residue (Complex Analysis)|... | By [[Existence of Laurent Series]], there exists a [[Definition:Laurent Series|Laurent series]]:
:$\ds \map f z = \sum_{n \mathop = -\infty}^\infty c_n \paren {z - a}^n$
convergent on $D \setminus \set a$.
As $f$ has a pole of order $N$ at $a$, we have $c_n = 0$ for $n < -N$.
So:
:$\ds \paren {z - a}^N \map f... | Residue at Multiple Pole | https://proofwiki.org/wiki/Residue_at_Multiple_Pole | https://proofwiki.org/wiki/Residue_at_Multiple_Pole | [
"Complex Analysis"
] | [
"Definition:Complex Function",
"Definition:Meromorphic Function",
"Definition:Region/Complex",
"Definition:Pole",
"Definition:Residue (Complex Analysis)"
] | [
"Existence of Laurent Series",
"Definition:Laurent Series",
"Definition:Taylor Series",
"Definition:Residue (Complex Analysis)",
"Taylor Series of Holomorphic Function",
"Category:Complex Analysis"
] |
proofwiki-15057 | Bijection from Cartesian Product of Initial Segments to Initial Segment | Let $\N_k$ be used to denote the set of the first $k$ non-zero natural numbers:
:$\N_k := \set {1, 2, \ldots, k}$
Then a bijection can be established between $\N_k \times \N_l$ and $\N_{k l}$, where $\N_k \times \N_l$ denotes the Cartesian product of $\N_k$ and $\N_l$. | Let $\phi: \N_k \times \N_l \to \N_{k l}$ be defined as:
:$\forall \tuple {m, n} \in \N_k \times \N_l: \map \phi {m, n} = \paren {m - 1} \times l + n$
First it is confirmed that the codomain of $\phi$ is indeed $\N_{k l}$.
{{finish|fiddly and tedious, can't think of an elegant way to prove it}} | Let $\N_k$ be used to denote the [[Definition:Set|set]] of the first $k$ [[Definition:Non-Zero Natural Number|non-zero natural numbers]]:
:$\N_k := \set {1, 2, \ldots, k}$
Then a [[Definition:Bijection|bijection]] can be established between $\N_k \times \N_l$ and $\N_{k l}$, where $\N_k \times \N_l$ denotes the [[De... | Let $\phi: \N_k \times \N_l \to \N_{k l}$ be defined as:
:$\forall \tuple {m, n} \in \N_k \times \N_l: \map \phi {m, n} = \paren {m - 1} \times l + n$
First it is confirmed that the [[Definition:Codomain of Mapping|codomain]] of $\phi$ is indeed $\N_{k l}$.
{{finish|fiddly and tedious, can't think of an elegant way... | Bijection from Cartesian Product of Initial Segments to Initial Segment | https://proofwiki.org/wiki/Bijection_from_Cartesian_Product_of_Initial_Segments_to_Initial_Segment | https://proofwiki.org/wiki/Bijection_from_Cartesian_Product_of_Initial_Segments_to_Initial_Segment | [
"Bijections",
"Natural Numbers"
] | [
"Definition:Set",
"Definition:Non-Zero Natural Number",
"Definition:Bijection",
"Definition:Cartesian Product"
] | [
"Definition:Codomain (Set Theory)/Mapping"
] |
proofwiki-15058 | Bijection between S x T and T x S | Let $S$ and $T$ be sets.
Let $S \times T$ be the Cartesian product of $S$ and $T$.
Then there exists a bijection from $S \times T$ to $T \times S$. | Let $\phi: S \times T \to T \times S$ be the mapping defined as:
:$\forall \tuple {s, t} \in S \times T: \map \phi {s, t} = \tuple {t, s}$
Then $\phi$ is the bijection required, as follows:
The domain of $\phi$ is $S \times T$.
Let $\tuple {t, s} \in T \times S$.
Then there exists $\tuple {s, t} \in S \times T$ such th... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $S \times T$ be the [[Definition:Cartesian Product|Cartesian product]] of $S$ and $T$.
Then there exists a [[Definition:Bijection|bijection]] from $S \times T$ to $T \times S$. | Let $\phi: S \times T \to T \times S$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall \tuple {s, t} \in S \times T: \map \phi {s, t} = \tuple {t, s}$
Then $\phi$ is the [[Definition:Bijection|bijection]] required, as follows:
The [[Definition:Domain of Mapping|domain]] of $\phi$ is $S \times T$.
Let $... | Bijection between S x T and T x S | https://proofwiki.org/wiki/Bijection_between_S_x_T_and_T_x_S | https://proofwiki.org/wiki/Bijection_between_S_x_T_and_T_x_S | [
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Bijection"
] | [
"Definition:Mapping",
"Definition:Bijection",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Surjection",
"Definition:Injection",
"Definition:Bijection"
] |
proofwiki-15059 | Bijection between R x (S x T) and (R x S) x T | Let $R$, $S$ and $T$ be sets.
Let $S \times T$ be the Cartesian product of $S$ and $T$.
Then there exists a bijection from $R \times \paren {S \times T}$ to $\paren {R \times S} \times T$.
Hence:
:$\card {R \times \paren {S \times T} } = \card {\paren {R \times S} \times T}$ | Let $\phi: R \times \paren {S \times T} \to \paren {R \times S} \times T$ be the mapping defined as:
:$\forall \tuple {r, \tuple {s, t} } \in R \times \paren {S \times T}: \map \phi {s, t} = \tuple {\tuple {r, s}, t}$
Then $\phi$ is the bijection required, as follows:
The domain of $\phi$ is $R \times \paren {S \times ... | Let $R$, $S$ and $T$ be [[Definition:Set|sets]].
Let $S \times T$ be the [[Definition:Cartesian Product|Cartesian product]] of $S$ and $T$.
Then there exists a [[Definition:Bijection|bijection]] from $R \times \paren {S \times T}$ to $\paren {R \times S} \times T$.
Hence:
:$\card {R \times \paren {S \times T} } = \... | Let $\phi: R \times \paren {S \times T} \to \paren {R \times S} \times T$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall \tuple {r, \tuple {s, t} } \in R \times \paren {S \times T}: \map \phi {s, t} = \tuple {\tuple {r, s}, t}$
Then $\phi$ is the [[Definition:Bijection|bijection]] required, as follows:
... | Bijection between R x (S x T) and (R x S) x T | https://proofwiki.org/wiki/Bijection_between_R_x_(S_x_T)_and_(R_x_S)_x_T | https://proofwiki.org/wiki/Bijection_between_R_x_(S_x_T)_and_(R_x_S)_x_T | [
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Bijection"
] | [
"Definition:Mapping",
"Definition:Bijection",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Surjection",
"Definition:Injection",
"Definition:Bijection"
] |
proofwiki-15060 | Bijection between Power Set of Disjoint Union and Cartesian Product of Power Sets | Let $S$ and $T$ be disjoint sets.
Let $\powerset S$ denote the power set of $S$.
Then there exists a bijection between $\powerset {S \cup T}$ and $\paren {\powerset S} \times \paren {\powerset T}$.
Hence:
:$\powerset {S \cup T} \sim \paren {\powerset S} \times \paren {\powerset T}$
where $\sim$ denotes set equivalence. | Let $\phi: \paren {\powerset S} \times \paren {\powerset T} \to \powerset {S \cup T}$ be defined as:
:$\forall \tuple {A, B} \in \paren {\powerset S} \times \paren {\powerset T}: \map \phi {A, B} = A \cup B$
In order to show that $\phi$ is a bijection, it needs to be shown that $\phi$ has the following properties:
:$\... | Let $S$ and $T$ be [[Definition:Disjoint Sets|disjoint sets]].
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Then there exists a [[Definition:Bijection|bijection]] between $\powerset {S \cup T}$ and $\paren {\powerset S} \times \paren {\powerset T}$.
Hence:
:$\powerset {S \cup T} \sim \pa... | Let $\phi: \paren {\powerset S} \times \paren {\powerset T} \to \powerset {S \cup T}$ be defined as:
:$\forall \tuple {A, B} \in \paren {\powerset S} \times \paren {\powerset T}: \map \phi {A, B} = A \cup B$
In order to show that $\phi$ is a [[Definition:Bijection|bijection]], it needs to be shown that $\phi$ has t... | Bijection between Power Set of Disjoint Union and Cartesian Product of Power Sets | https://proofwiki.org/wiki/Bijection_between_Power_Set_of_Disjoint_Union_and_Cartesian_Product_of_Power_Sets | https://proofwiki.org/wiki/Bijection_between_Power_Set_of_Disjoint_Union_and_Cartesian_Product_of_Power_Sets | [
"Power Set",
"Set Union",
"Cartesian Product"
] | [
"Definition:Disjoint Sets",
"Definition:Power Set",
"Definition:Bijection",
"Definition:Set Equivalence"
] | [
"Definition:Bijection",
"Definition:Property",
"Definition:Mapping",
"Definition:Left-Total Relation",
"Definition:Many-to-One Relation",
"Definition:Surjection",
"Definition:Injection",
"Definition:Left-Total Relation",
"Definition:Many-to-One Relation",
"Definition:Mapping",
"Definition:Power ... |
proofwiki-15061 | Bijection between Power Set of nth Initial Section and Initial Section of nth Power of 2 | Let $\N_n$ be used to denote the first $n$ non-zero natural numbers:
:$\N_n = \set {1, 2, \ldots, n}$
Then there exists a bijection between the power set of $\N_n$ and $\N_{2^n}$. | Let $\phi: \powerset {\N_n} \to \N_{2^n}$ be defined as:
:$\forall A \in \powerset {\N_n}: \map \phi A = \begin{cases} \ds \sum_{k \mathop \in A} 2^{k - 1} & : A \ne \O \\ 2^k & : A = \O \end{cases}$
Apart from $\O$, every $A \in \powerset {\N_n}$ consists of a set of integers between $1$ and $n$.
The expression $\ds \... | Let $\N_n$ be used to denote the [[Definition:Initial Segment of One-Based Natural Numbers|first $n$ non-zero natural numbers]]:
:$\N_n = \set {1, 2, \ldots, n}$
Then there exists a [[Definition:Bijection|bijection]] between the [[Definition:Power Set|power set]] of $\N_n$ and $\N_{2^n}$. | Let $\phi: \powerset {\N_n} \to \N_{2^n}$ be defined as:
:$\forall A \in \powerset {\N_n}: \map \phi A = \begin{cases} \ds \sum_{k \mathop \in A} 2^{k - 1} & : A \ne \O \\ 2^k & : A = \O \end{cases}$
Apart from $\O$, every $A \in \powerset {\N_n}$ consists of a [[Definition:Set|set]] of [[Definition:Integer|integers]... | Bijection between Power Set of nth Initial Section and Initial Section of nth Power of 2 | https://proofwiki.org/wiki/Bijection_between_Power_Set_of_nth_Initial_Section_and_Initial_Section_of_nth_Power_of_2 | https://proofwiki.org/wiki/Bijection_between_Power_Set_of_nth_Initial_Section_and_Initial_Section_of_nth_Power_of_2 | [
"Power Set",
"Integer Powers"
] | [
"Definition:Initial Segment of Natural Numbers/One-Based",
"Definition:Bijection",
"Definition:Power Set"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Summation",
"Definition:Set",
"Definition:Power (Algebra)/Integer",
"Definition:Left-Total Relation",
"Definition:Many-to-One Relation",
"Definition:Mapping",
"Basis Representation Theorem",
"Definition:Integer",
"Definition:Addition/Sum",
"D... |
proofwiki-15062 | Trivial Group is Group | The trivial group is a group. | Let $G = \struct {\set e, \circ}$ be an algebraic structure. | The [[Definition:Trivial Group|trivial group]] is a [[Definition:Group|group]]. | Let $G = \struct {\set e, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]]. | Trivial Group is Group | https://proofwiki.org/wiki/Trivial_Group_is_Group | https://proofwiki.org/wiki/Trivial_Group_is_Group | [
"Trivial Group"
] | [
"Definition:Trivial Group",
"Definition:Group"
] | [
"Definition:Algebraic Structure/One Operation"
] |
proofwiki-15063 | Symmetry Group of Line Segment is Group | The symmetry group of the line segment is a group. | Let us refer to this group as $D_1$.
Taking the group axioms in turn: | The [[Definition:Symmetry Group of Line Segment|symmetry group of the line segment]] is a [[Definition:Group|group]]. | Let us refer to this group as $D_1$.
Taking the [[Axiom:Group Axioms|group axioms]] in turn: | Symmetry Group of Line Segment is Group | https://proofwiki.org/wiki/Symmetry_Group_of_Line_Segment_is_Group | https://proofwiki.org/wiki/Symmetry_Group_of_Line_Segment_is_Group | [
"Symmetry Group of Line Segment"
] | [
"Definition:Symmetry Group of Line Segment",
"Definition:Group",
"Definition:Symmetry Group of Line Segment",
"Definition:Symmetry Group of Line Segment"
] | [
"Axiom:Group Axioms"
] |
proofwiki-15064 | Order of Dihedral Group | The dihedral group $D_n$ is of order $2 n$. | By definition, $D_n$ is the symmetry group of the regular polygon of $n$ sides.
:500px 500px
Let $P$ be a regular $n$-gon.
By inspection, it is seen that:
:$(1): \quad$ there are $n$ symmetries of the vertices of $P$ by rotation
:$(2): \quad$ there are a further $n$ symmetries of the vertices of $P$ by rotation after r... | The [[Definition:Dihedral Group|dihedral group]] $D_n$ is of [[Definition:Order of Structure|order]] $2 n$. | By definition, $D_n$ is the [[Definition:Symmetry Group|symmetry group]] of the [[Definition:Regular Polygon|regular polygon]] of $n$ [[Definition:Side of Polygon|sides]].
:[[File:SymmetryGroupOddPolygon.png|500px]] [[File:SymmetryGroupEvenPolygon.png|500px]]
Let $P$ be a [[Definition:Regular Polygon|regular $n$-gon]... | Order of Dihedral Group | https://proofwiki.org/wiki/Order_of_Dihedral_Group | https://proofwiki.org/wiki/Order_of_Dihedral_Group | [
"Dihedral Groups"
] | [
"Definition:Dihedral Group",
"Definition:Order of Structure"
] | [
"Definition:Symmetry Group",
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"File:SymmetryGroupOddPolygon.png",
"File:SymmetryGroupEvenPolygon.png",
"Definition:Polygon/Regular",
"Definition:Symmetry (Geometry)",
"Definition:Polygon/Vertex",
"Definition:Rotation (Geometry)/Plane",
"Defin... |
proofwiki-15065 | Klein Four-Group is Group | The Klein $4$-group $K_4$ is a group. | From Klein Four-Group as Subgroup of $S_4$ it is demonstrated that $K_4$ is a subgroup of the $4$th symmetric group.
Hence the result.
{{qed}} | The [[Definition:Klein Four-Group|Klein $4$-group]] $K_4$ is a [[Definition:Group|group]]. | From [[Klein Four-Group as Subgroup of S4|Klein Four-Group as Subgroup of $S_4$]] it is demonstrated that $K_4$ is a [[Definition:Subgroup|subgroup]] of the [[Definition:Symmetric Group|$4$th symmetric group]].
Hence the result.
{{qed}} | Klein Four-Group is Group | https://proofwiki.org/wiki/Klein_Four-Group_is_Group | https://proofwiki.org/wiki/Klein_Four-Group_is_Group | [
"Klein Four-Group"
] | [
"Definition:Klein Four-Group",
"Definition:Group"
] | [
"Klein Four-Group as Subgroup of S4",
"Definition:Subgroup",
"Definition:Symmetric Group"
] |
proofwiki-15066 | Dihedral Group is Non-Abelian | Let $n \in \N$ be a natural number such that $n > 2$.
Let $D_n$ denote the dihedral group of order $2 n$.
Then $D_n$ is not abelian. | From Group of Order less than 6 is Abelian we have that $D_1$ and $D_2$ are abelian, which is why the condition on $n$.
From Group Presentation of Dihedral Group we have:
:$\beta \alpha = \alpha^{n - 1} \beta$
for some $\alpha, \beta \in D_n$ such that $\alpha \ne \beta$.
We also have:
:$\alpha^n = e$
But if $D_n$ were... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n > 2$.
Let $D_n$ denote the [[Definition:Dihedral Group|dihedral group]] of [[Definition:Order of Structure|order]] $2 n$.
Then $D_n$ is not [[Definition:Abelian Group|abelian]]. | From [[Group of Order less than 6 is Abelian]] we have that $D_1$ and $D_2$ are [[Definition:Abelian Group|abelian]], which is why the condition on $n$.
From [[Group Presentation of Dihedral Group]] we have:
:$\beta \alpha = \alpha^{n - 1} \beta$
for some $\alpha, \beta \in D_n$ such that $\alpha \ne \beta$.
We also ... | Dihedral Group is Non-Abelian | https://proofwiki.org/wiki/Dihedral_Group_is_Non-Abelian | https://proofwiki.org/wiki/Dihedral_Group_is_Non-Abelian | [
"Dihedral Groups"
] | [
"Definition:Natural Numbers",
"Definition:Dihedral Group",
"Definition:Order of Structure",
"Definition:Abelian Group"
] | [
"Group of Order less than 6 is Abelian",
"Definition:Abelian Group",
"Dihedral Group/Group Presentation",
"Definition:Abelian Group",
"Cancellation Laws",
"Definition:Dihedral Group"
] |
proofwiki-15067 | Matrix Entrywise Addition is Commutative | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is commutative on $\map \MM {m, n}$.
That is:
:$\mathbf A + \... | From:
:Integers form Ring
:Rational Numbers form Ring
:Real Numbers form Ring
:Complex Numbers form Ring
the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings.
Hence we can apply Matrix Entrywise Addition over Ring is Commutative.
{{qed|lemma}}
The above cannot be applied to the natural numbers $\N$, as they ... | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition|matrix entrywise sum]] of ... | From:
:[[Integers form Ring]]
:[[Rational Numbers form Ring]]
:[[Real Numbers form Ring]]
:[[Complex Numbers form Ring]]
the [[Definition:Standard Number System|standard number systems]] $\Z$, $\Q$, $\R$ and $\C$ are [[Definition:Ring (Abstract Algebra)|rings]].
Hence we can apply [[Matrix Entrywise Addition over Rin... | Matrix Entrywise Addition is Commutative/Proof 1 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Commutative | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Commutative/Proof_1 | [
"Matrix Entrywise Addition is Commutative",
"Matrix Entrywise Addition",
"Examples of Commutative Operations"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Matrix Entrywise Addition",
"Definition:Commutative/Operation"
] | [
"Integers form Commutative Ring",
"Rational Numbers form Ring",
"Real Numbers form Ring",
"Complex Numbers form Ring",
"Definition:Number",
"Definition:Ring (Abstract Algebra)",
"Matrix Entrywise Addition over Ring is Commutative",
"Definition:Natural Numbers",
"Definition:Ring (Abstract Algebra)",
... |
proofwiki-15068 | Matrix Entrywise Addition is Commutative | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is commutative on $\map \MM {m, n}$.
That is:
:$\mathbf A + \... | Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be matrices whose order is $m \times n$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf B
| r = \sqbrk a_{m n} + \sqbrk b_{m n}
| c = Definition of $\mathbf A$ and $\mathbf B$
}}
{{eqn | r = \sqbrk {a + b}_{m n}
| c = {{Defof|Matri... | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition|matrix entrywise sum]] of ... | Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be [[Definition:Matrix|matrices]] whose [[Definition:Order of Matrix|order]] is $m \times n$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf B
| r = \sqbrk a_{m n} + \sqbrk b_{m n}
| c = Definition of $\mathbf A$ and $\mathbf B$
}}
{{e... | Matrix Entrywise Addition is Commutative/Proof 2 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Commutative | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Commutative/Proof_2 | [
"Matrix Entrywise Addition is Commutative",
"Matrix Entrywise Addition",
"Examples of Commutative Operations"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Matrix Entrywise Addition",
"Definition:Commutative/Operation"
] | [
"Definition:Matrix",
"Definition:Matrix/Order",
"Commutative Law of Addition"
] |
proofwiki-15069 | Special Linear Group is not Abelian | Let $K$ be a field whose zero is $0_K$ and unity is $1_K$.
Let $\SL {n, K}$ be the special linear group of order $n$ over $K$.
Then $\SL {n, K}$ is not an abelian group. | From Special Linear Group is Subgroup of General Linear Group we have that $\SL {n, K}$ is a group.
From Matrix Multiplication is not Commutative it follows that $\SL {n, K}$ is not abelian.
{{qed}} | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_K$ and [[Definition:Unity of Field|unity]] is $1_K$.
Let $\SL {n, K}$ be the [[Definition:Special Linear Group|special linear group of order $n$ over $K$]].
Then $\SL {n, K}$ is not an [[Definition:Abelian Group|abe... | From [[Special Linear Group is Subgroup of General Linear Group]] we have that $\SL {n, K}$ is a [[Definition:Group|group]].
From [[Matrix Multiplication is not Commutative]] it follows that $\SL {n, K}$ is not [[Definition:Abelian Group|abelian]].
{{qed}} | Special Linear Group is not Abelian | https://proofwiki.org/wiki/Special_Linear_Group_is_not_Abelian | https://proofwiki.org/wiki/Special_Linear_Group_is_not_Abelian | [
"Special Linear Group"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Special Linear Group",
"Definition:Abelian Group"
] | [
"Special Linear Group is Subgroup of General Linear Group",
"Definition:Group",
"Matrix Multiplication is not Commutative",
"Definition:Abelian Group"
] |
proofwiki-15070 | Summation Formula (Complex Analysis) | :$\ds \sum_{n \mathop \in \Z \mathop \setminus X} \map f n = - \sum_{z_0 \mathop \in X} \Res {\pi \map \cot {\pi z} \map f z} {z_0}$ | By Summation Formula: Lemma, there exists a constant $A$ such that:
:$\cmod {\map \cot {\pi z} } < A$
for all $z$ on $C_N$.
Since $f$ has only finitely many poles, we can take $N$ large enough so that no poles of $f$ lie on $C_N$.
Let $X_N$ be the set of poles of $f$ contained in the region bounded by $C_N$.
From P... | :$\ds \sum_{n \mathop \in \Z \mathop \setminus X} \map f n = - \sum_{z_0 \mathop \in X} \Res {\pi \map \cot {\pi z} \map f z} {z_0}$ | By [[Summation Formula (Complex Analysis)/Lemma|Summation Formula: Lemma]], there exists a constant $A$ such that:
:$\cmod {\map \cot {\pi z} } < A$
for all $z$ on $C_N$.
Since $f$ has only [[Definition:Finite Set|finitely many]] [[Definition:Pole (Complex Analysis)|poles]], we can take $N$ large enough so that n... | Summation Formula (Complex Analysis) | https://proofwiki.org/wiki/Summation_Formula_(Complex_Analysis) | https://proofwiki.org/wiki/Summation_Formula_(Complex_Analysis) | [
"Complex Analysis"
] | [] | [
"Summation Formula (Complex Analysis)/Lemma",
"Definition:Finite Set",
"Definition:Isolated Singularity/Pole",
"Definition:Isolated Singularity/Pole",
"Definition:Set",
"Definition:Isolated Singularity/Pole",
"Definition:Region/Complex",
"Definition:Boundary (Geometry)",
"Poles of Cotangent Function... |
proofwiki-15071 | Summation Formula (Complex Analysis)/Lemma | Let $N \in \N$ be an arbitrary natural number.
Let $C_N$ be the square embedded in the complex plane with vertices $\paren {N + \dfrac 1 2} \paren {\pm 1 \pm i}$.
Then there exists a constant real number $A$ independent of $N$ such that:
:$\cmod {\map \cot {\pi z} } < A$
for all $z \in C_N$. | Let $z = x + iy$ for real $x, y$. | Let $N \in \N$ be an arbitrary [[Definition:Natural Number|natural number]].
Let $C_N$ be the [[Definition:Square (Geometry)|square]] embedded in the [[Definition:Complex Plane|complex plane]] with [[Definition:Vertex of Polygon|vertices]] $\paren {N + \dfrac 1 2} \paren {\pm 1 \pm i}$.
Then there exists a [[Definit... | Let $z = x + iy$ for real $x, y$. | Summation Formula (Complex Analysis)/Lemma | https://proofwiki.org/wiki/Summation_Formula_(Complex_Analysis)/Lemma | https://proofwiki.org/wiki/Summation_Formula_(Complex_Analysis)/Lemma | [
"Complex Analysis"
] | [
"Definition:Natural Numbers",
"Definition:Quadrilateral/Square",
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Constant",
"Definition:Real Number"
] | [] |
proofwiki-15072 | Product of Generating Elements of Dihedral Group | Let $D_n$ be the dihedral group of order $2 n$.
Let $D_n$ be defined by its group presentation:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Then for all $k \in \Z_{\ge 0}$:
:$\beta \alpha^k = \alpha^{n - k} \beta$ | The proof proceeds by induction.
For all $k \in \Z_{\ge 0}$, let $\map P k$ be the proposition:
:$\beta \alpha^k = \alpha^{n - k} \beta$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \beta \alpha^0
| r = \beta e
| c =
}}
{{eqn | r = e \beta
| c =
}}
{{eqn | r = \alpha^n \beta
| c =
}}
{{e... | Let $D_n$ be the [[Definition:Dihedral Group|dihedral group]] of [[Definition:Order of Structure|order]] $2 n$.
Let $D_n$ be defined by its [[Group Presentation of Dihedral Group|group presentation]]:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Then for all $k \in \Z_{\ge ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $k \in \Z_{\ge 0}$, let $\map P k$ be the [[Definition:Proposition|proposition]]:
:$\beta \alpha^k = \alpha^{n - k} \beta$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \beta \alpha^0
| r = \beta e
| c =
}}
{{eqn | r =... | Product of Generating Elements of Dihedral Group | https://proofwiki.org/wiki/Product_of_Generating_Elements_of_Dihedral_Group | https://proofwiki.org/wiki/Product_of_Generating_Elements_of_Dihedral_Group | [
"Dihedral Groups"
] | [
"Definition:Dihedral Group",
"Definition:Order of Structure",
"Dihedral Group/Group Presentation"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-15073 | Center of Dihedral Group | Let $n \in \N$ be a natural number such that $n \ge 3$.
Let $D_n$ be the dihedral group of order $2 n$, given by:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Let $\map Z {D_n}$ denote the center of $D_n$.
Then:
:$\map Z {D_n} = \begin{cases} e & : n \text { odd} \\ \set {e, ... | By definition, the center of $D_n$ is:
:$\map Z {D_n} = \set {g \in D_n: g x = x g, \forall x \in D_n}$
For $n \le 2$ we have that $\order {D_n} \le 4$ and so by Group of Order less than 6 is Abelian $D_n$ is abelian for $n < 3$.
Hence by definition of abelian group:
:$\map Z {D_n} = D_n$
for $n < 3$.
So, let $n \ge 3$... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n \ge 3$.
Let $D_n$ be the [[Definition:Dihedral Group|dihedral group]] of [[Definition:Order of Structure|order]] $2 n$, given by:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Let $\map Z {D_n}$ ... | By definition, the [[Definition:Center of Group|center]] of $D_n$ is:
:$\map Z {D_n} = \set {g \in D_n: g x = x g, \forall x \in D_n}$
For $n \le 2$ we have that $\order {D_n} \le 4$ and so by [[Group of Order less than 6 is Abelian]] $D_n$ is [[Definition:Abelian Group|abelian]] for $n < 3$.
Hence by definition of [... | Center of Dihedral Group | https://proofwiki.org/wiki/Center_of_Dihedral_Group | https://proofwiki.org/wiki/Center_of_Dihedral_Group | [
"Dihedral Groups",
"Centers of Groups"
] | [
"Definition:Natural Numbers",
"Definition:Dihedral Group",
"Definition:Order of Structure",
"Definition:Center (Abstract Algebra)/Group"
] | [
"Definition:Center (Abstract Algebra)/Group",
"Group of Order less than 6 is Abelian",
"Definition:Abelian Group",
"Definition:Abelian Group",
"Dihedral Group/Group Presentation",
"Product of Generating Elements of Dihedral Group",
"Definition:Generator of Group",
"Definition:Order of Group Element",
... |
proofwiki-15074 | Intersection of Additive Groups of Integer Multiples | Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.
Let $\struct {m \Z, +}$ and $\struct {n \Z, +}$ be the corresponding additive groups of integer multiples.
Then:
:$\struct {m \Z, +} \cap \struct {n \Z, +} = \struct {\lcm \set {m, n} \Z, +}$ | By definition:
:$m \Z = \set {x \in \Z: m \divides x}$
Thus:
{{begin-eqn}}
{{eqn | l = m \Z \cap n \Z
| r = \set {x \in \Z: n \divides x} \cap \set {x \in \Z: m \divides x}
| c =
}}
{{eqn | r = \set {x \in \Z: n \divides x \land m \divides x}
| c =
}}
{{eqn | r = \set {x \in \Z: \lcm \set {m, n} \di... | Let $m, n \in \Z_{> 0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\struct {m \Z, +}$ and $\struct {n \Z, +}$ be the corresponding [[Definition:Additive Group of Integer Multiples|additive groups of integer multiples]].
Then:
:$\struct {m \Z, +} \cap \struct {n \Z, +} = \struct {\... | By definition:
:$m \Z = \set {x \in \Z: m \divides x}$
Thus:
{{begin-eqn}}
{{eqn | l = m \Z \cap n \Z
| r = \set {x \in \Z: n \divides x} \cap \set {x \in \Z: m \divides x}
| c =
}}
{{eqn | r = \set {x \in \Z: n \divides x \land m \divides x}
| c =
}}
{{eqn | r = \set {x \in \Z: \lcm \set {m, n} \... | Intersection of Additive Groups of Integer Multiples | https://proofwiki.org/wiki/Intersection_of_Additive_Groups_of_Integer_Multiples | https://proofwiki.org/wiki/Intersection_of_Additive_Groups_of_Integer_Multiples | [
"Additive Groups of Integer Multiples"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Additive Group of Integer Multiples"
] | [] |
proofwiki-15075 | Subgroups of Additive Group of Integers Modulo m | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$.
The subgroups of $\struct {\Z_m, +_m}$ are the additive groups of integers modulo $k$ where:
:$k \divides m$ | From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is cyclic.
Let $H$ be a subgroup of $\struct {\Z_m, +_m}$
From Subgroup of Cyclic Group is Cyclic, $H$ is of the form $\struct {\Z_k, +_k}$ for some $k \in \Z$.
From Lagrange's Theorem, it follows that $k \divides m$.
Hence the result.
{{qed... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\struct {\Z_m, +_m}$ denote the [[Definition:Additive Group of Integers|additive group of integers modulo $m$]].
The [[Definition:Subgroup|subgroups]] of $\struct {\Z_m, +_m}$ are the [[Definition:Additive Group of In... | From [[Integers Modulo m under Addition form Cyclic Group]], $\struct {\Z_m, +_m}$ is [[Definition:Cyclic Group|cyclic]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $\struct {\Z_m, +_m}$
From [[Subgroup of Cyclic Group is Cyclic]], $H$ is of the form $\struct {\Z_k, +_k}$ for some $k \in \Z$.
From [[Lagrange's... | Subgroups of Additive Group of Integers Modulo m | https://proofwiki.org/wiki/Subgroups_of_Additive_Group_of_Integers_Modulo_m | https://proofwiki.org/wiki/Subgroups_of_Additive_Group_of_Integers_Modulo_m | [
"Additive Groups of Integers Modulo m"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Additive Group of Integers",
"Definition:Subgroup",
"Definition:Additive Group of Integers Modulo m"
] | [
"Integers Modulo m under Addition form Cyclic Group",
"Definition:Cyclic Group",
"Definition:Subgroup",
"Subgroup of Cyclic Group is Cyclic",
"Lagrange's Theorem (Group Theory)"
] |
proofwiki-15076 | Subgroup of Additive Group of Integers Generated by Two Integers | Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.
Let $\struct {\Z, +}$ denote the additive group of integers.
Let $\gen {m, n}$ be the subgroup of $\struct {\Z, +}$ generated by $m$ and $n$.
Then:
:$\gen {m, n} = \struct {\gcd \set {m, n} \Z, +}$
That is, the additive groups of integer multiples of $\gcd \set {... | By definition:
:$\gen {m, n} = \set {x \in \Z: \gcd \set {m, n} \divides x}$
{{Handwaving|Sorry, I would make the effort, but it's tedious.}}
Hence the result.
{{qed}} | Let $m, n \in \Z_{> 0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\struct {\Z, +}$ denote the [[Definition:Additive Group of Integers|additive group of integers]].
Let $\gen {m, n}$ be the [[Definition:Subgroup|subgroup]] of $\struct {\Z, +}$ [[Definition:Generator of Subgroup|gen... | By definition:
:$\gen {m, n} = \set {x \in \Z: \gcd \set {m, n} \divides x}$
{{Handwaving|Sorry, I would make the effort, but it's tedious.}}
Hence the result.
{{qed}} | Subgroup of Additive Group of Integers Generated by Two Integers | https://proofwiki.org/wiki/Subgroup_of_Additive_Group_of_Integers_Generated_by_Two_Integers | https://proofwiki.org/wiki/Subgroup_of_Additive_Group_of_Integers_Generated_by_Two_Integers | [
"Additive Groups of Integer Multiples"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Additive Group of Integers",
"Definition:Subgroup",
"Definition:Generator of Subgroup",
"Definition:Additive Group of Integer Multiples",
"Definition:Greatest Common Divisor/Integers"
] | [] |
proofwiki-15077 | Subgroups of Cartesian Product of Additive Group of Integers | Let $\struct {\Z, +}$ denote the additive group of integers.
Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.
Let $\struct {\Z \times \Z, +}$ denote the Cartesian product of $\struct {\Z, +}$ with itself.
The subgroups of $\struct {\Z \times \Z, +}$ are not all of the form:
:$\struct {m \Z, +} \times \struct {n... | Consider the map $\phi: \struct {m \Z, +} \times \struct {n \Z, +} \mapsto \struct {\Z, +} \times \struct {\Z, +}$ defined by:
:$\forall c, d \in \Z: \map \phi {m c, n d} = \tuple {c, d}$
which is a group isomorphism.
{{explain|Prove the above statement}}
Hence, $\struct {m \Z, +} \times \struct {n \Z, +}$ is a free ab... | Let $\struct {\Z, +}$ denote the [[Definition:Additive Group of Integers|additive group of integers]].
Let $m, n \in \Z_{> 0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\struct {\Z \times \Z, +}$ denote the [[Definition:Cartesian Product|Cartesian product]] of $\struct {\Z, +}$ wi... | Consider the map $\phi: \struct {m \Z, +} \times \struct {n \Z, +} \mapsto \struct {\Z, +} \times \struct {\Z, +}$ defined by:
:$\forall c, d \in \Z: \map \phi {m c, n d} = \tuple {c, d}$
which is a [[Definition:Group Isomorphism|group isomorphism]].
{{explain|Prove the above statement}}
Hence, $\struct {m \Z, +} \t... | Subgroups of Cartesian Product of Additive Group of Integers | https://proofwiki.org/wiki/Subgroups_of_Cartesian_Product_of_Additive_Group_of_Integers | https://proofwiki.org/wiki/Subgroups_of_Cartesian_Product_of_Additive_Group_of_Integers | [
"Additive Group of Integers",
"Additive Groups of Integer Multiples"
] | [
"Definition:Additive Group of Integers",
"Definition:Strictly Positive/Integer",
"Definition:Cartesian Product",
"Definition:Subgroup",
"Definition:Additive Group of Integer Multiples"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Free Abelian Group on Set",
"Definition:Rank (Free Abelian Group)",
"Definition:Subgroup",
"Definition:Generator of Subgroup",
"Definition:Singleton",
"Definition:Subgroup"
] |
proofwiki-15078 | Equivalence Relation on Symmetric Group by Image of n is Congruence Modulo Subgroup | Let $S_n$ denote the symmetric group on $n$ letters $\set {1, \dots, n}$.
Let $\sim$ be the relation on $S_n$ defined as:
:$\forall \pi, \tau \in S_n: \pi \sim \tau \iff \map \pi n = \map \tau n$
Then $\sim$ is an equivalence relation which is congruence modulo a subgroup.
{{explain|Work needed to be done to explain ex... | We claim that $\sim$ is left congruence modulo $S_{n - 1}$, the symmetric group on $n - 1$ letters $\set {1, \dots, n - 1}$.
Notice that every element of $S_{n - 1}$ fixes $n$.
For all $\pi, \tau \in S_n$ such that $\pi \sim \tau$:
{{begin-eqn}}
{{eqn | l = \map {\paren {\pi^{-1} \circ \tau} } n
| r = \map {\pi^{... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]] $\set {1, \dots, n}$.
Let $\sim$ be the [[Definition:Relation|relation on $S_n$]] defined as:
:$\forall \pi, \tau \in S_n: \pi \sim \tau \iff \map \pi n = \map \tau n$
Then $\sim$ is an [[Definition:Equivalence Relation|eq... | We claim that $\sim$ is [[Definition:Left Congruence Modulo Subgroup|left congruence modulo $S_{n - 1}$]], the [[Definition:Symmetric Group on n Letters|symmetric group on $n - 1$ letters]] $\set {1, \dots, n - 1}$.
Notice that every element of $S_{n - 1}$ fixes $n$.
For all $\pi, \tau \in S_n$ such that $\pi \sim \... | Equivalence Relation on Symmetric Group by Image of n is Congruence Modulo Subgroup | https://proofwiki.org/wiki/Equivalence_Relation_on_Symmetric_Group_by_Image_of_n_is_Congruence_Modulo_Subgroup | https://proofwiki.org/wiki/Equivalence_Relation_on_Symmetric_Group_by_Image_of_n_is_Congruence_Modulo_Subgroup | [
"Symmetric Groups"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Relation",
"Definition:Equivalence Relation",
"Definition:Congruence Modulo Subgroup"
] | [
"Definition:Congruence Modulo Subgroup/Left Congruence",
"Definition:Symmetric Group/n Letters",
"Definition:Congruence Modulo Subgroup/Left Congruence",
"Definition:Congruence Modulo Subgroup/Left Congruence",
"Definition:Equivalence Relation",
"Left Congruence Modulo Subgroup is Equivalence Relation"
] |
proofwiki-15079 | Order of Additive Group of Integers Modulo m | Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$.
The order of $\struct {\Z_m, +_m}$ is $m$. | By definition, the order of a group is the cardinality of its underlying set.
By definition, the underlying set of $\struct {\Z_m, +_m}$ is the set of residue classes $\Z_m$:
:$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$
From Cardinality of Set of Residue Classes, $\Z_m$ has $m$ elements.
Henc... | Let $\struct {\Z_m, +_m}$ denote the [[Definition:Additive Group of Integers Modulo m|additive group of integers modulo $m$]].
The [[Definition:Order of Group|order]] of $\struct {\Z_m, +_m}$ is $m$. | By definition, the [[Definition:Order of Group|order]] of a [[Definition:Group|group]] is the [[Definition:Cardinality|cardinality]] of its [[Definition:Underlying Set of Structure|underlying set]].
By definition, the [[Definition:Underlying Set of Structure|underlying set]] of $\struct {\Z_m, +_m}$ is the [[Definitio... | Order of Additive Group of Integers Modulo m | https://proofwiki.org/wiki/Order_of_Additive_Group_of_Integers_Modulo_m | https://proofwiki.org/wiki/Order_of_Additive_Group_of_Integers_Modulo_m | [
"Additive Groups of Integers Modulo m"
] | [
"Definition:Additive Group of Integers Modulo m",
"Definition:Order of Structure"
] | [
"Definition:Order of Structure",
"Definition:Group",
"Definition:Cardinality",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Set of Residue Classes",
"Cardinality of Set of Residue Classes",
"Definition:Element"
] |
proofwiki-15080 | Order of Multiplicative Group of Reduced Residues | Let $\struct {\Z'_m, \times_m}$ denote the multiplicative group of reduced residues modulo $m$.
The order of $\struct {\Z'_m, \times_m}$ is $\map \phi m$, where $\phi$ denotes the Euler $\phi$ function. | By definition, the order of a group is the cardinality of its underlying set.
By definition, the underlying set of $\struct {\Z'_m, \times_m}$ is the reduced residue system $\Z'_m$:
:$\Z'_m = \set {\eqclass {a_1} m, \eqclass {a_2} m, \ldots, \eqclass {a_{\map \phi m} } m}$
where:
:$\forall k: a_k \perp m$
From Cardina... | Let $\struct {\Z'_m, \times_m}$ denote the [[Definition:Multiplicative Group of Reduced Residues|multiplicative group of reduced residues modulo $m$]].
The [[Definition:Order of Group|order]] of $\struct {\Z'_m, \times_m}$ is $\map \phi m$, where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function... | By definition, the [[Definition:Order of Group|order]] of a [[Definition:Group|group]] is the [[Definition:Cardinality|cardinality]] of its [[Definition:Underlying Set of Structure|underlying set]].
By definition, the [[Definition:Underlying Set of Structure|underlying set]] of $\struct {\Z'_m, \times_m}$ is the [[Def... | Order of Multiplicative Group of Reduced Residues | https://proofwiki.org/wiki/Order_of_Multiplicative_Group_of_Reduced_Residues | https://proofwiki.org/wiki/Order_of_Multiplicative_Group_of_Reduced_Residues | [
"Multiplicative Groups of Reduced Residues"
] | [
"Definition:Multiplicative Group of Reduced Residues",
"Definition:Order of Structure",
"Definition:Euler Phi Function"
] | [
"Definition:Order of Structure",
"Definition:Group",
"Definition:Cardinality",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Reduced Residue System",
"Cardinality of Reduced Residue System",
"Definition:Element"
] |
proofwiki-15081 | Circle Group is Uncountably Infinite | The circle group $\struct {K, \times}$ is an uncountably infinite group. | From Quotient Group of Reals by Integers is Circle Group, $\struct {K, \times}$ is isomorphic to the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$.
But $\dfrac {\struct {\R, +} } {\struct {\Z, +} }$ is the half-open interval $\hointr 0 1$.
A real interval is uncountable by Real Numbers are Uncountable.
{{exp... | The [[Definition:Circle Group|circle group]] $\struct {K, \times}$ is an [[Definition:Uncountable Set|uncountably]] [[Definition:Infinite Group|infinite group]]. | From [[Quotient Group of Reals by Integers is Circle Group]], $\struct {K, \times}$ is [[Definition:Group Isomorphism|isomorphic]] to the [[Definition:Quotient Group|quotient group]] of $\struct {\R, +}$ by $\struct {\Z, +}$.
But $\dfrac {\struct {\R, +} } {\struct {\Z, +} }$ is the [[Definition:Half-Open Real Interva... | Circle Group is Uncountably Infinite | https://proofwiki.org/wiki/Circle_Group_is_Uncountably_Infinite | https://proofwiki.org/wiki/Circle_Group_is_Uncountably_Infinite | [
"Circle Group",
"Uncountable Sets"
] | [
"Definition:Circle Group",
"Definition:Uncountable/Set",
"Definition:Infinite Group"
] | [
"Quotient Group of Reals by Integers is Circle Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Quotient Group",
"Definition:Real Interval/Half-Open",
"Definition:Real Interval",
"Definition:Uncountable/Set",
"Real Numbers are Uncountably Infinite",
"Real Numbers are ... |
proofwiki-15082 | Subgroup of Symmetric Group that Fixes n | Let $S_n$ denote the symmetric group on $n$ letters.
Let $H$ denote the subgroup of $S_n$ which consists of all $\pi \in S_n$ such that:
:$\map \pi n = n$
Then:
:$H = S_{n - 1}$
and the index of $H$ in $S_n$ is given by:
:$\index {S_n} H = n$ | We have that $S_{n - 1}$ is the symmetric group on $n - 1$ letters.
Let $\pi \in S_{n - 1}$.
Then $\pi$ is a permutation on $n - 1$ letters.
Hence $\pi$ is also a permutation on $n$ letters which fixes $n$.
So $S_{n - 1} \subseteq H$.
Now let $\pi \in H$.
Then $\pi$ is a permutation on $n$ letters which fixes $n$.
That... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $H$ denote the [[Definition:Subgroup|subgroup]] of $S_n$ which consists of all $\pi \in S_n$ such that:
:$\map \pi n = n$
Then:
:$H = S_{n - 1}$
and the [[Definition:Index of Subgroup|index]] of $H$ in $S_n$ is give... | We have that $S_{n - 1}$ is the [[Definition:Symmetric Group on n Letters|symmetric group on $n - 1$ letters]].
Let $\pi \in S_{n - 1}$.
Then $\pi$ is a [[Definition:Permutation on n Letters|permutation on $n - 1$ letters]].
Hence $\pi$ is also a [[Definition:Permutation on n Letters|permutation on $n$ letters]] wh... | Subgroup of Symmetric Group that Fixes n | https://proofwiki.org/wiki/Subgroup_of_Symmetric_Group_that_Fixes_n | https://proofwiki.org/wiki/Subgroup_of_Symmetric_Group_that_Fixes_n | [
"Symmetric Groups"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Subgroup",
"Definition:Index of Subgroup"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Permutation on n Letters",
"Definition:Permutation on n Letters",
"Definition:Fixed Element under Permutation",
"Definition:Permutation on n Letters",
"Definition:Fixed Element under Permutation",
"Definition:Permutation on n Letters",
"Definition:Gr... |
proofwiki-15083 | Order of Product of Abelian Group Elements Divides LCM of Orders of Elements | Let $G$ be an abelian group.
Let $a, b \in G$.
Then:
:$\order {a b} \divides \lcm \set {\order a, \order b}$
where:
:$\order a$ denotes the order of $a$
:$\divides$ denotes divisibility
:$\lcm$ denotes the lowest common multiple. | Let $\order a = m, \order b = n$.
Let $c = \lcm \set {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = c
| r = r m
| c = for some $r \in \Z$
}}
{{eqn | r = s n
| c = for some $s \in \Z$
}}
{{end-eqn}}
So:
{{begin-eqn}}
{{eqn | l = \paren {a b}^c
| r = a^c b^c
| c = Power of Product of Commuting Ele... | Let $G$ be an [[Definition:Abelian Group|abelian group]].
Let $a, b \in G$.
Then:
:$\order {a b} \divides \lcm \set {\order a, \order b}$
where:
:$\order a$ denotes the [[Definition:Order of Group Element|order]] of $a$
:$\divides$ denotes [[Definition:Divisor of Integer|divisibility]]
:$\lcm$ denotes the [[Definitio... | Let $\order a = m, \order b = n$.
Let $c = \lcm \set {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = c
| r = r m
| c = for some $r \in \Z$
}}
{{eqn | r = s n
| c = for some $s \in \Z$
}}
{{end-eqn}}
So:
{{begin-eqn}}
{{eqn | l = \paren {a b}^c
| r = a^c b^c
| c = [[Power of Product of Commu... | Order of Product of Abelian Group Elements Divides LCM of Orders of Elements | https://proofwiki.org/wiki/Order_of_Product_of_Abelian_Group_Elements_Divides_LCM_of_Orders_of_Elements | https://proofwiki.org/wiki/Order_of_Product_of_Abelian_Group_Elements_Divides_LCM_of_Orders_of_Elements | [
"Order of Group Elements",
"Abelian Groups"
] | [
"Definition:Abelian Group",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Lowest Common Multiple/Integers"
] | [
"Power of Product of Commuting Elements in Semigroup equals Product of Powers",
"Element to Power of Multiple of Order is Identity"
] |
proofwiki-15084 | Elements of Abelian Group whose Order Divides n is Subgroup | Let $G$ be an abelian group whose identity element is $e$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer .
Let $G_n$ be the subset of $G$ defined as:
:$G_n = \set {x \in G: \order x \divides n}$
where:
:$\order x$ denotes the order of $x$
:$\divides$ denotes divisibility.
Then $G_n$ is a subgroup of $G$. | From Identity is Only Group Element of Order 1:
:$\order e = 1$
and so from One Divides all Integers:
:$\order e \divides n$
Thus $G_n \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = G_n
| c =
}}
{{eqn | ll= \leadsto
| l = \order x
| o = \divides
| r = n
| c =
}}
{... | Let $G$ be an [[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]] .
Let $G_n$ be the [[Definition:Subset|subset]] of $G$ defined as:
:$G_n = \set {x \in G: \order x \div... | From [[Identity is Only Group Element of Order 1]]:
:$\order e = 1$
and so from [[One Divides all Integers]]:
:$\order e \divides n$
Thus $G_n \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = G_n
| c =
}}
{{eqn | ll= \leadsto
| l = \order x
| o = \divides
| r = n
... | Elements of Abelian Group whose Order Divides n is Subgroup | https://proofwiki.org/wiki/Elements_of_Abelian_Group_whose_Order_Divides_n_is_Subgroup | https://proofwiki.org/wiki/Elements_of_Abelian_Group_whose_Order_Divides_n_is_Subgroup | [
"Order of Group Elements",
"Subgroups",
"Abelian Groups"
] | [
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Strictly Positive/Integer",
"Definition:Subset",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subgroup"
] | [
"Identity is Only Group Element of Order 1",
"Integer Divisor Results/One Divides all Integers",
"Order of Group Element equals Order of Inverse",
"Order of Product of Abelian Group Elements Divides LCM of Orders of Elements",
"Definition:Lowest Common Multiple/Integers",
"Two-Step Subgroup Test"
] |
proofwiki-15085 | Groups of Order 4 | There exist exactly $2$ groups of order $4$, up to isomorphism:
:$C_4$, the cyclic group of order $4$
:$K_4$, the Klein $4$-group. | From Existence of Cyclic Group of Order n we have that one such group of order $4$ is the cyclic group of order $4$:
This is exemplified by the additive group of integers modulo $4$, whose Cayley table can be presented as:
{{:Modulo Addition/Cayley Table/Modulo 4}}
From Group whose Order equals Order of Element is Cycl... | There exist exactly $2$ [[Definition:Group|groups]] of [[Definition:Order of Group|order]] $4$, up to [[Definition:Group Isomorphism|isomorphism]]:
:$C_4$, the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $4$
:$K_4$, the [[Definition:Klein Four-Group|Klein $4$-group]]. | From [[Existence of Cyclic Group of Order n]] we have that one such [[Definition:Group|group]] of [[Definition:Order of Group|order]] $4$ is the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $4$:
This is exemplified by the [[Definition:Additive Group of Integers Modulo m|additive grou... | Groups of Order 4 | https://proofwiki.org/wiki/Groups_of_Order_4 | https://proofwiki.org/wiki/Groups_of_Order_4 | [
"Groups of Order 4"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Klein Four-Group"
] | [
"Existence of Cyclic Group of Order n",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Additive Group of Integers Modulo m",
"Modulo Addition/Cayley Table/Modulo 4",
"Group whose Order equals Order of Element is Cyclic",
"... |
proofwiki-15086 | Groups of Order 6 | There exist exactly $2$ groups of order $6$, up to isomorphism:
:$C_6$, the cyclic group of order $6$
:$S_3$, the symmetric group on $3$ letters. | From Existence of Cyclic Group of Order n we have that one such group of order $6$ is $C_6$ the cyclic group of order $6$:
This is exemplified by the additive group of integers modulo $6$, whose Cayley table can be presented as:
{{:Modulo Addition/Cayley Table/Modulo 6}}
Then we have the symmetric group on $3$ letters.... | There exist exactly $2$ [[Definition:Group|groups]] of [[Definition:Order of Group|order]] $6$, up to [[Definition:Group Isomorphism|isomorphism]]:
:$C_6$, the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $6$
:$S_3$, the [[Definition:Symmetric Group on n Letters|symmetric group on $... | From [[Existence of Cyclic Group of Order n]] we have that one such [[Definition:Group|group]] of [[Definition:Order of Group|order]] $6$ is $C_6$ the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $6$:
This is exemplified by the [[Definition:Additive Group of Integers Modulo m|additiv... | Groups of Order 6 | https://proofwiki.org/wiki/Groups_of_Order_6 | https://proofwiki.org/wiki/Groups_of_Order_6 | [
"Groups of Order 6"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Symmetric Group/n Letters"
] | [
"Existence of Cyclic Group of Order n",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Additive Group of Integers Modulo m",
"Modulo Addition/Cayley Table/Modulo 6",
"Definition:Symmetric Group/n Letters",
"Order of Symmet... |
proofwiki-15087 | Finite Group whose Subsets form Chain is Cyclic P-Group | Let $G$ be a group.
Let $G$ be such that its subgroups form a chain.
Then $G$ is a cyclic $p$-group. | Suppose $G$ is not a $p$-group.
Then there exist two distinct primes $p_1, p_2$.
By Cauchy's Group Theorem, there exist subgroups $H, K$ such that:
:$\order H = p_1$
:$\order K = p_2$
That is:
:$H = \gen a$
:$k = \gen b$
for some $a, b \in G: a \ne b$, where:
:$\order a = p_1$
:$\order b = p_2$
and so both $H \nsubsete... | Let $G$ be a [[Definition:Group|group]].
Let $G$ be such that its [[Definition:Subgroup|subgroups]] form a [[Definition:Chain of Sets|chain]].
Then $G$ is a [[Definition:Cyclic Group|cyclic]] [[Definition:P-Group|$p$-group]]. | Suppose $G$ is not a [[Definition:P-Group|$p$-group]].
Then there exist two [[Definition:Distinct Elements|distinct]] [[Definition:Prime Number|primes]] $p_1, p_2$.
By [[Cauchy's Group Theorem]], there exist [[Definition:Subgroup|subgroups]] $H, K$ such that:
:$\order H = p_1$
:$\order K = p_2$
That is:
:$H = \gen a... | Finite Group whose Subsets form Chain is Cyclic P-Group | https://proofwiki.org/wiki/Finite_Group_whose_Subsets_form_Chain_is_Cyclic_P-Group | https://proofwiki.org/wiki/Finite_Group_whose_Subsets_form_Chain_is_Cyclic_P-Group | [
"Cyclic Groups",
"P-Groups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Cyclic Group",
"Definition:P-Group"
] | [
"Definition:P-Group",
"Definition:Distinct/Plural",
"Definition:Prime Number",
"Cauchy's Group Theorem",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Chain (Order Theory)/Subset Relation",
"Rule of Transposition",
"Definition:P-Group",
"Definition:Cyclic Group",
"Definition:Cyclic G... |
proofwiki-15088 | Subgroup of Circle Group Generated by Distinct Roots of Unity | Let $K$ be the circle group.
Let $m, n \in \Z_{>0}$ be (strictly) positive integers.
Let $d = \lcm \set {m, n}$ be the least common multiple of $m$ and $n$.
Let $\alpha$ be a primitive $n$th root of unity.
Let $\beta$ be a primitive $m$th root of unity.
Let $\gamma$ be a primitive $d$th root of unity.
Let $H = \gen {\a... | {{ProofWanted|This is (probably) a specialisation of a more general result on cyclic groups.}} | Let $K$ be the [[Definition:Circle Group|circle group]].
Let $m, n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $d = \lcm \set {m, n}$ be the [[Definition:Least Common Multiple|least common multiple]] of $m$ and $n$.
Let $\alpha$ be a [[Definition:Primitive Complex Root... | {{ProofWanted|This is (probably) a specialisation of a more general result on cyclic groups.}} | Subgroup of Circle Group Generated by Distinct Roots of Unity | https://proofwiki.org/wiki/Subgroup_of_Circle_Group_Generated_by_Distinct_Roots_of_Unity | https://proofwiki.org/wiki/Subgroup_of_Circle_Group_Generated_by_Distinct_Roots_of_Unity | [
"Multiplicative Groups of Complex Roots of Unity",
"Circle Group"
] | [
"Definition:Circle Group",
"Definition:Strictly Positive/Integer",
"Definition:Lowest Common Multiple",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Subgroup",
"Definition:Generator of Subgroup"
] | [] |
proofwiki-15089 | Multiplicative Group of Complex Roots of Unity is Subgroup of Circle Group | Let $n \in \Z$ be an integer such that $n > 0$.
Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity.
Let $\struct {K, \times}$ denote the circle group.
Then $\struct {U_n, \times}$ is a subgroup of $\struct {K, \times}$. | By definition of the multiplicative group of complex $n$th roots of unity:
:$U_n := \set {z \in \C: z^n = 1}$
By definition of the circle group:
:$K = \set {z \in \C: \cmod z = 1}$
By Modulus of Complex Root of Unity equals 1:
:$\forall z \in U_n: \cmod z = 1$
Thus:
:$U_n \subseteq K$
We further have that the operation... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n > 0$.
Let $\struct {U_n, \times}$ denote the [[Definition:Multiplicative Group of Complex Roots of Unity|multiplicative group of complex $n$th roots of unity]].
Let $\struct {K, \times}$ denote the [[Definition:Circle Group|circle group]].
Then $\stru... | By definition of the [[Definition:Multiplicative Group of Complex Roots of Unity|multiplicative group of complex $n$th roots of unity]]:
:$U_n := \set {z \in \C: z^n = 1}$
By definition of the [[Definition:Circle Group|circle group]]:
:$K = \set {z \in \C: \cmod z = 1}$
By [[Modulus of Complex Root of Unity equals 1]... | Multiplicative Group of Complex Roots of Unity is Subgroup of Circle Group | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Roots_of_Unity_is_Subgroup_of_Circle_Group | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Roots_of_Unity_is_Subgroup_of_Circle_Group | [
"Circle Group",
"Multiplicative Groups of Complex Roots of Unity"
] | [
"Definition:Integer",
"Definition:Multiplicative Group of Complex Roots of Unity",
"Definition:Circle Group",
"Definition:Subgroup"
] | [
"Definition:Multiplicative Group of Complex Roots of Unity",
"Definition:Circle Group",
"Modulus of Complex Root of Unity equals 1",
"Definition:Operation/Binary Operation",
"Definition:Multiplication/Complex Numbers",
"Roots of Unity under Multiplication form Cyclic Group",
"Definition:Group",
"Defin... |
proofwiki-15090 | Intersection of Multiplicative Groups of Complex Roots of Unity | Let $\struct {K, \times}$ denote the circle group.
Let $m, n \in \Z_{>0}$ be (strictly) positive integers.
Let $c = \lcm \set {m, n}$ be the lowest common multiple of $m$ and $n$.
Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity.
Let $\struct {U_m, \times}$ denote the multipli... | {{ProofWanted|This is (probably) a specialisation of a more general result on cyclic groups.}} | Let $\struct {K, \times}$ denote the [[Definition:Circle Group|circle group]].
Let $m, n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $c = \lcm \set {m, n}$ be the [[Definition:Lowest Common Multiple of Integers|lowest common multiple]] of $m$ and $n$.
Let $\struct {U_n... | {{ProofWanted|This is (probably) a specialisation of a more general result on cyclic groups.}} | Intersection of Multiplicative Groups of Complex Roots of Unity | https://proofwiki.org/wiki/Intersection_of_Multiplicative_Groups_of_Complex_Roots_of_Unity | https://proofwiki.org/wiki/Intersection_of_Multiplicative_Groups_of_Complex_Roots_of_Unity | [
"Multiplicative Groups of Complex Roots of Unity",
"Circle Group"
] | [
"Definition:Circle Group",
"Definition:Strictly Positive/Integer",
"Definition:Lowest Common Multiple/Integers",
"Definition:Multiplicative Group of Complex Roots of Unity",
"Definition:Multiplicative Group of Complex Roots of Unity"
] | [] |
proofwiki-15091 | Direct Product of Normal Subgroups is Normal | Let $G$ and $G'$ be groups.
Let:
:$H \lhd G$
:$H' \lhd G'$
where $\lhd$ denotes the relation of being a normal subgroup.
Then:
:$\paren {H \times H'} \lhd \paren {G \times G'}$
where $H \times H'$ denotes the group direct product of $H$ and $H'$ | Let $\tuple {x, x'} \in G \times G'$ and $\tuple {y, y'} \in H \times H'$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, x'} \tuple {y, y'} \tuple {x, x'}^{-1}
| r = \tuple {x, x'} \tuple {y, y'} \tuple {x^{-1}, x'^{-1} }
| c =
}}
{{eqn | r = \tuple {x y x^{-1}, x' y' x'^{-1} }
| c =
}}
{{eqn | o = \in... | Let $G$ and $G'$ be [[Definition:Group|groups]].
Let:
:$H \lhd G$
:$H' \lhd G'$
where $\lhd$ denotes the relation of being a [[Definition:Normal Subgroup|normal subgroup]].
Then:
:$\paren {H \times H'} \lhd \paren {G \times G'}$
where $H \times H'$ denotes the [[Definition:Group Direct Product|group direct product]... | Let $\tuple {x, x'} \in G \times G'$ and $\tuple {y, y'} \in H \times H'$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, x'} \tuple {y, y'} \tuple {x, x'}^{-1}
| r = \tuple {x, x'} \tuple {y, y'} \tuple {x^{-1}, x'^{-1} }
| c =
}}
{{eqn | r = \tuple {x y x^{-1}, x' y' x'^{-1} }
| c =
}}
{{eqn | o = \i... | Direct Product of Normal Subgroups is Normal | https://proofwiki.org/wiki/Direct_Product_of_Normal_Subgroups_is_Normal | https://proofwiki.org/wiki/Direct_Product_of_Normal_Subgroups_is_Normal | [
"Group Direct Products",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Group Direct Product"
] | [] |
proofwiki-15092 | Normalizer of Rotation in Dihedral Group | Let $n \in \N$ be a natural number such that $n \ge 3$.
Let $D_n$ be the dihedral group of order $2 n$, given by:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Let $\map {N_{D_n} } {\set \alpha}$ denote the normalizer of the singleton containing the rotation element $\alpha$.
... | By definition, the normalizer of $\set \alpha$ is:
:$\map {N_{D_n} } {\set \alpha} := \set {g \in D_n: g \set \alpha g^{-1} = \set \alpha}$
That is:
:$\map {N_{D_n} } {\set \alpha} := \set {g \in D_n: g \alpha = \alpha g}$
First let $g = \alpha^k$ for some $k \in \Z$.
Then:
:$\alpha \alpha^k = \alpha^k \alpha$
which in... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n \ge 3$.
Let $D_n$ be the [[Definition:Dihedral Group|dihedral group]] of [[Definition:Order of Structure|order]] $2 n$, given by:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Let $\map {N_{D_n} ... | By definition, the [[Definition:Normalizer|normalizer]] of $\set \alpha$ is:
:$\map {N_{D_n} } {\set \alpha} := \set {g \in D_n: g \set \alpha g^{-1} = \set \alpha}$
That is:
:$\map {N_{D_n} } {\set \alpha} := \set {g \in D_n: g \alpha = \alpha g}$
First let $g = \alpha^k$ for some $k \in \Z$.
Then:
:$\alpha \alph... | Normalizer of Rotation in Dihedral Group | https://proofwiki.org/wiki/Normalizer_of_Rotation_in_Dihedral_Group | https://proofwiki.org/wiki/Normalizer_of_Rotation_in_Dihedral_Group | [
"Dihedral Groups",
"Normalizers"
] | [
"Definition:Natural Numbers",
"Definition:Dihedral Group",
"Definition:Order of Structure",
"Definition:Normalizer",
"Definition:Singleton",
"Definition:Rotation (Geometry)/Plane",
"Definition:Generated Subgroup"
] | [
"Definition:Normalizer",
"Cancellation Laws",
"Definition:Generator of Subgroup",
"Category:Dihedral Groups",
"Category:Normalizers"
] |
proofwiki-15093 | Normalizer of Reflection in Dihedral Group | Let $n \in \N$ be a natural number such that $n \ge 3$.
Let $D_n$ be the dihedral group of order $2 n$, given by:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Let $\map {N_{D_n} } {\set \beta}$ denote the normalizer of the singleton containing the reflection element $\beta$.
... | By definition, the normalizer of $\set \beta$ is:
:$\map {N_{D_n} } {\set \beta} := \set {g \in D_n: g \set \beta g^{-1} = \set \beta}$
That is:
:$\map {N_{D_n} } {\set \beta} := \set {g \in D_n: g \beta = \beta g}$
First let $g = \beta^k$ for $k \in \set {0, 1}$.
Then:
:$\beta \beta^k = \beta^k \beta$
Thus:
:$\forall ... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n \ge 3$.
Let $D_n$ be the [[Definition:Dihedral Group|dihedral group]] of [[Definition:Order of Structure|order]] $2 n$, given by:
:$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$
Let $\map {N_{D_n} ... | By definition, the [[Definition:Normalizer|normalizer]] of $\set \beta$ is:
:$\map {N_{D_n} } {\set \beta} := \set {g \in D_n: g \set \beta g^{-1} = \set \beta}$
That is:
:$\map {N_{D_n} } {\set \beta} := \set {g \in D_n: g \beta = \beta g}$
First let $g = \beta^k$ for $k \in \set {0, 1}$.
Then:
:$\beta \beta^k = ... | Normalizer of Reflection in Dihedral Group | https://proofwiki.org/wiki/Normalizer_of_Reflection_in_Dihedral_Group | https://proofwiki.org/wiki/Normalizer_of_Reflection_in_Dihedral_Group | [
"Dihedral Groups",
"Normalizers"
] | [
"Definition:Natural Numbers",
"Definition:Dihedral Group",
"Definition:Order of Structure",
"Definition:Normalizer",
"Definition:Singleton",
"Definition:Reflection (Geometry)/Plane"
] | [
"Definition:Normalizer",
"Cancellation Laws",
"Product of Generating Elements of Dihedral Group",
"Definition:Odd Integer",
"Definition:Even Integer",
"Category:Dihedral Groups",
"Category:Normalizers"
] |
proofwiki-15094 | Normalizer of Subgroup of Symmetric Group that Fixes n | Let $S_n$ denote the symmetric group on $n$ letters.
Let $H$ denote the subgroup of $S_n$ which consists of all $\pi \in S_n$ such that:
:$\map \pi n = n$
The normalizer of $H$ is given by:
:$\map {N_{S_n} } H = \map {N_{S_n} } {S_{n - 1} } = S_{n - 1}$ | We have from Subgroup of Symmetric Group that Fixes n that $N = S_{n - 1}$.
By definition of normalizer:
:$\map {N_{S_n} } {S_{n - 1} } := \set {\rho \in S_n: \rho S_{n - 1} \rho^{-1} = S_{n - 1} }$
We have from Group is Normal in Itself that:
:$\forall \rho \in S_{n - 1}: \rho S_{n - 1} \rho^{-1} \in S_{n - 1}$
and so... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $H$ denote the [[Definition:Subgroup|subgroup]] of $S_n$ which consists of all $\pi \in S_n$ such that:
:$\map \pi n = n$
The [[Definition:Normalizer|normalizer]] of $H$ is given by:
:$\map {N_{S_n} } H = \map {N_{S... | We have from [[Subgroup of Symmetric Group that Fixes n]] that $N = S_{n - 1}$.
By definition of [[Definition:Normalizer|normalizer]]:
:$\map {N_{S_n} } {S_{n - 1} } := \set {\rho \in S_n: \rho S_{n - 1} \rho^{-1} = S_{n - 1} }$
We have from [[Group is Normal in Itself]] that:
:$\forall \rho \in S_{n - 1}: \rho S_... | Normalizer of Subgroup of Symmetric Group that Fixes n | https://proofwiki.org/wiki/Normalizer_of_Subgroup_of_Symmetric_Group_that_Fixes_n | https://proofwiki.org/wiki/Normalizer_of_Subgroup_of_Symmetric_Group_that_Fixes_n | [
"Symmetric Groups",
"Normalizers"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Subgroup",
"Definition:Normalizer"
] | [
"Subgroup of Symmetric Group that Fixes n",
"Definition:Normalizer",
"Group is Normal in Itself",
"Definition:Set Difference",
"Definition:Permutation on n Letters",
"Definition:Bijection",
"Definition:Fixed Element under Permutation",
"Intersection with Complement is Empty iff Subset",
"Definition:... |
proofwiki-15095 | Center of Quaternion Group | Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group.
Let $\map Z {\Dic 2}$ denote the center of $\Dic 2$.
Then:
:$\map Z {\Dic 2} = \set {e, a^2}$ | By definition, the center of $\Dic 2$ is:
:$\map Z {\Dic 2} = \set {g \in \Dic 2: g x = x g, \forall x \in \Dic 2}$
We are given that:
:$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$
We have that $\Dic 2$ is generated by $\alpha$ and $\beta$.
Thus:
:$x \in \map Z {\Dic 2} \iff x a = a x \land x b = b x$
Let $x \... | Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the [[Definition:Quaternion Group|quaternion group]].
Let $\map Z {\Dic 2}$ denote the [[Definition:Center of Group|center of $\Dic 2$]].
Then:
:$\map Z {\Dic 2} = \set {e, a^2}$ | By definition, the [[Definition:Center of Group|center]] of $\Dic 2$ is:
:$\map Z {\Dic 2} = \set {g \in \Dic 2: g x = x g, \forall x \in \Dic 2}$
We are given that:
:$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$
We have that $\Dic 2$ is [[Definition:Generator of Group|generated]] by $\alpha$ and $\beta$.
... | Center of Quaternion Group | https://proofwiki.org/wiki/Center_of_Quaternion_Group | https://proofwiki.org/wiki/Center_of_Quaternion_Group | [
"Quaternion Group",
"Centers of Groups"
] | [
"Definition:Dicyclic Group/Quaternion Group",
"Definition:Center (Abstract Algebra)/Group"
] | [
"Definition:Center (Abstract Algebra)/Group",
"Definition:Generator of Group",
"Definition:Order of Group Element",
"Product of Generating Elements of Quaternion Group"
] |
proofwiki-15096 | Product of Generating Elements of Quaternion Group | Let $Q = \Dic 2$ be the quaternion group:
:$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$
Then for all $k \in \Z_{\ge 0}$:
:$b a^k = a^{-k} b$ | The proof proceeds by induction.
For all $k \in \Z_{\ge 0}$, let $\map P k$ be the proposition:
:$b a^k = a^{-k} b$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = b a^0
| r = b e
| c =
}}
{{eqn | r = e b
| c =
}}
{{eqn | r = a^{-0} b
| c =
}}
{{end-eqn}}
Thus $\map P 0$ is seen to hold. | Let $Q = \Dic 2$ be the [[Definition:Quaternion Group|quaternion group]]:
:$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$
Then for all $k \in \Z_{\ge 0}$:
:$b a^k = a^{-k} b$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $k \in \Z_{\ge 0}$, let $\map P k$ be the [[Definition:Proposition|proposition]]:
:$b a^k = a^{-k} b$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = b a^0
| r = b e
| c =
}}
{{eqn | r = e b
| c =
}}
{{eqn | r = a... | Product of Generating Elements of Quaternion Group | https://proofwiki.org/wiki/Product_of_Generating_Elements_of_Quaternion_Group | https://proofwiki.org/wiki/Product_of_Generating_Elements_of_Quaternion_Group | [
"Quaternion Group"
] | [
"Definition:Dicyclic Group/Quaternion Group"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-15097 | Conjugacy Classes of Quaternion Group | Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group.
The conjugacy classes of $\Dic 2$ are:
:$\set e, \set {a^2}, \set {a, a^3}, \set {b, a^2 b}, \set {a b, a^3 b}$ | From Center of Quaternion Group, we have:
:$\map Z {\Dic 2} = \set {e, a^2}$
Thus from Conjugacy Class of Element of Center is Singleton, $\set e$ and $\set {a^2}$ are two of those conjugacy classes.
By inspection of the Cayley table:
{{:Quaternion Group/Cayley Table}}
we investigate the remaining $6$ elements in turn,... | Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the [[Definition:Quaternion Group|quaternion group]].
The [[Definition:Conjugacy Class|conjugacy classes]] of $\Dic 2$ are:
:$\set e, \set {a^2}, \set {a, a^3}, \set {b, a^2 b}, \set {a b, a^3 b}$ | From [[Center of Quaternion Group]], we have:
:$\map Z {\Dic 2} = \set {e, a^2}$
Thus from [[Conjugacy Class of Element of Center is Singleton]], $\set e$ and $\set {a^2}$ are two of those [[Definition:Conjugacy Class|conjugacy classes]].
By inspection of the [[Quaternion Group/Cayley Table|Cayley table]]:
{{:Quater... | Conjugacy Classes of Quaternion Group | https://proofwiki.org/wiki/Conjugacy_Classes_of_Quaternion_Group | https://proofwiki.org/wiki/Conjugacy_Classes_of_Quaternion_Group | [
"Quaternion Group",
"Examples of Conjugacy Classes"
] | [
"Definition:Dicyclic Group/Quaternion Group",
"Definition:Conjugacy Class"
] | [
"Center of Quaternion Group",
"Conjugacy Class of Element of Center is Singleton",
"Definition:Conjugacy Class",
"Quaternion Group/Cayley Table",
"Definition:Conjugacy Class",
"Definition:Conjugacy Class",
"Definition:Conjugacy Class"
] |
proofwiki-15098 | Conjugacy Action on Subsets is Group Action | Let $\powerset G$ be the set of all subgroups of $G$.
For any $S \in \powerset G$ and for any $g \in G$, the conjugacy action:
:$g * S := g \circ S \circ g^{-1}$
is a group action. | By {{Group-axiom|2}}, $e \in G$, thus:
{{begin-eqn}}
{{eqn | l = e * S
| r = e \circ S \circ e^{-1}
| c = Definition of $*$
}}
{{eqn | r = S
| c = {{Defof|Identity Element}}
}}
{{end-eqn}}
Thus {{GroupActionAxiom|2}} is seen to be fulfilled.
Then:
{{begin-eqn}}
{{eqn | l = \paren {g_1 \circ g_2} * S
... | Let $\powerset G$ be the set of all [[Definition:Subgroup|subgroups]] of $G$.
For any $S \in \powerset G$ and for any $g \in G$, the [[Definition:Conjugacy Action on Subsets|conjugacy action]]:
:$g * S := g \circ S \circ g^{-1}$
is a [[Definition:Group Action|group action]]. | By {{Group-axiom|2}}, $e \in G$, thus:
{{begin-eqn}}
{{eqn | l = e * S
| r = e \circ S \circ e^{-1}
| c = Definition of $*$
}}
{{eqn | r = S
| c = {{Defof|Identity Element}}
}}
{{end-eqn}}
Thus {{GroupActionAxiom|2}} is seen to be fulfilled.
Then:
{{begin-eqn}}
{{eqn | l = \paren {g_1 \circ g_2}... | Conjugacy Action on Subsets is Group Action | https://proofwiki.org/wiki/Conjugacy_Action_on_Subsets_is_Group_Action | https://proofwiki.org/wiki/Conjugacy_Action_on_Subsets_is_Group_Action | [
"Conjugacy Action"
] | [
"Definition:Subgroup",
"Definition:Conjugacy Action/Subsets",
"Definition:Group Action"
] | [
"Inverse of Group Product"
] |
proofwiki-15099 | Normed Vector Space Requires Multiplicative Norm on Division Ring | Let $R$ be a normed division ring with a submultiplicative norm $\norm {\, \cdot \,}_R$.
Let $V$ be a vector space that is not a trivial vector space.
Let $\norm {\, \cdot \,}: V \to \R_{\ge 0}$ be a mapping from $V$ to the positive real numbers satisfying the vector space norm axioms.
Then $\norm {\, \cdot \,}_R$ is a... | Since $V$ is not a trivial vector space:
:$\exists \mathbf v \in V: \mathbf v \ne 0$
By {{NormAxiomVector|1}}:
:$\norm {\mathbf v} > 0$
Let $r, s \in R$:
{{begin-eqn}}
{{eqn | l = \norm {r s}_R \norm {\mathbf v}
| r = \norm {\paren {r s} \mathbf v}
| c = {{NormAxiomVector|2}}
}}
{{eqn | r = \norm {r \paren ... | Let $R$ be a [[Definition:Normed Division Ring|normed division ring]] with a [[Definition:Submultiplicative Norm on Ring|submultiplicative norm]] $\norm {\, \cdot \,}_R$.
Let $V$ be a [[Definition:Vector Space|vector space]] that is not a [[Definition:Trivial Vector Space|trivial vector space]].
Let $\norm {\, \cdot ... | Since $V$ is not a [[Definition:Trivial Vector Space|trivial vector space]]:
:$\exists \mathbf v \in V: \mathbf v \ne 0$
By {{NormAxiomVector|1}}:
:$\norm {\mathbf v} > 0$
Let $r, s \in R$:
{{begin-eqn}}
{{eqn | l = \norm {r s}_R \norm {\mathbf v}
| r = \norm {\paren {r s} \mathbf v}
| c = {{NormAxiomV... | Normed Vector Space Requires Multiplicative Norm on Division Ring | https://proofwiki.org/wiki/Normed_Vector_Space_Requires_Multiplicative_Norm_on_Division_Ring | https://proofwiki.org/wiki/Normed_Vector_Space_Requires_Multiplicative_Norm_on_Division_Ring | [
"Normed Division Rings",
"Norm Theory"
] | [
"Definition:Normed Division Ring",
"Definition:Norm/Ring/Submultiplicative",
"Definition:Vector Space",
"Definition:Trivial Vector Space",
"Definition:Positive/Real Number",
"Axiom:Vector Space Norm Axioms",
"Definition:Norm/Ring/Multiplicative"
] | [
"Definition:Trivial Vector Space",
"Category:Normed Division Rings",
"Category:Norm Theory"
] |
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