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-15200 | Normality Relation is not Transitive | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $K$ be a normal subgroup of $N$.
Then it is not necessarily the case that $K$ is a normal subgroup of $G$. | Proof by Counterexample:
Let $D_4$ denote the dihedral group $D_4$.
Let $D_4$ be presented in matrix representation:
{{:Dihedral Group D4/Matrix Representation/Formulation 1}}
Its Cayley table is given as:
{{:Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table}}
Consider the subgroup $H$ whose underlying... | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $K$ be a [[Definition:Normal Subgroup|normal subgroup]] of $N$.
Then it is not necessarily the case that $K$ is a [[Definition:Normal Subgroup|normal subgroup]] of $G$. | [[Proof by Counterexample]]:
Let $D_4$ denote the [[Definition:Dihedral Group D4|dihedral group $D_4$]].
Let $D_4$ be presented in [[Dihedral Group D4/Matrix Representation/Formulation 1|matrix representation]]:
{{:Dihedral Group D4/Matrix Representation/Formulation 1}}
Its [[Dihedral Group D4/Matrix Representation... | Normality Relation is not Transitive/Proof 2 | https://proofwiki.org/wiki/Normality_Relation_is_not_Transitive | https://proofwiki.org/wiki/Normality_Relation_is_not_Transitive/Proof_2 | [
"Normal Subgroups",
"Normality Relation is not Transitive"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup"
] | [
"Proof by Counterexample",
"Definition:Dihedral Group D4",
"Dihedral Group D4/Matrix Representation/Formulation 1",
"Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table",
"Definition:Subgroup",
"Definition:Underlying Set/Abstract Algebra",
"Subgroup of Index 2 is Normal",
"Definition:No... |
proofwiki-15201 | Stabilizer is Normal iff Stabilizer of Each Element of Orbit | Let $\struct {G, \circ}$ be a group.
Let $S$ be a set.
Let $*: G \times S \to S$ be a group action.
Let $x \in S$.
Let $\Stab x$ denote the stabilizer of $x$ under $*$.
Let $\Orb x$ denote the orbit of $x$ under $*$.
Then $\Stab x$ is normal in $G$ {{iff}} $\Stab x$ is also the stabilizer of every element in $\Orb x$. | {{tidy}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S$ be a [[Definition:Set|set]].
Let $*: G \times S \to S$ be a [[Definition:Group Action|group action]].
Let $x \in S$.
Let $\Stab x$ denote the [[Definition:Stabilizer|stabilizer]] of $x$ under $*$.
Let $\Orb x$ denote the [[Definition:Orbit (Group Th... | {{tidy}} | Stabilizer is Normal iff Stabilizer of Each Element of Orbit | https://proofwiki.org/wiki/Stabilizer_is_Normal_iff_Stabilizer_of_Each_Element_of_Orbit | https://proofwiki.org/wiki/Stabilizer_is_Normal_iff_Stabilizer_of_Each_Element_of_Orbit | [
"Stabilizers",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Group Action",
"Definition:Stabilizer",
"Definition:Orbit (Group Theory)",
"Definition:Normal Subgroup",
"Definition:Stabilizer",
"Definition:Element"
] | [] |
proofwiki-15202 | Power of Coset Product is Coset of Power | Let $\struct {G, \circ}$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $a \in G$.
Then:
:$\forall n \in \Z: \paren {a \circ N}^n = \paren {a^n} \circ N$ | From Quotient Group is Group, the operation:
:$\forall a, b \in G: \paren {a \circ N} \circ \paren {b \circ N} = \paren {a \circ b} \circ N$
is the group operation in the quotient group $\struct {G / N, \circ}$.
The result follows directly by definition of power of group element.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $a \in G$.
Then:
:$\forall n \in \Z: \paren {a \circ N}^n = \paren {a^n} \circ N$ | From [[Quotient Group is Group]], the operation:
:$\forall a, b \in G: \paren {a \circ N} \circ \paren {b \circ N} = \paren {a \circ b} \circ N$
is the [[Definition:Group Operation|group operation]] in the [[Definition:Quotient Group|quotient group]] $\struct {G / N, \circ}$.
The result follows directly by definition... | Power of Coset Product is Coset of Power | https://proofwiki.org/wiki/Power_of_Coset_Product_is_Coset_of_Power | https://proofwiki.org/wiki/Power_of_Coset_Product_is_Coset_of_Power | [
"Coset Product"
] | [
"Definition:Group",
"Definition:Normal Subgroup"
] | [
"Quotient Group is Group",
"Definition:Group Product/Group Law",
"Definition:Quotient Group",
"Definition:Power of Element/Group"
] |
proofwiki-15203 | Condition for Power of Element of Quotient Group to be Identity | Let $G$ be a group whose identity is $e$.
Let $N$ be a normal subgroup of $G$.
Let $a \in G$.
Then:
:$\paren {a N}^n$ is the identity of the quotient group $G / N$
{{iff}}:
:$a^n \in N$ | Let $\paren {a N}^n$ be the identity of $G / N$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {a N}^n
| r = N
| c = Quotient Group is Group: {{Group-axiom|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {a^n} N
| r = N
| c = Power of Coset Product is Coset of Power
}}
{{eqn | ll= \leadstoandfrom... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $a \in G$.
Then:
:$\paren {a N}^n$ is the [[Definition:Identity Element|identity]] of the [[Definition:Quotient Group|quotient group]] $G / N$
{{iff}... | Let $\paren {a N}^n$ be the [[Definition:Identity Element|identity]] of $G / N$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {a N}^n
| r = N
| c = [[Quotient Group is Group]]: {{Group-axiom|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {a^n} N
| r = N
| c = [[Power of Coset Product is Coset ... | Condition for Power of Element of Quotient Group to be Identity | https://proofwiki.org/wiki/Condition_for_Power_of_Element_of_Quotient_Group_to_be_Identity | https://proofwiki.org/wiki/Condition_for_Power_of_Element_of_Quotient_Group_to_be_Identity | [
"Coset Product"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Normal Subgroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Quotient Group"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Quotient Group is Group",
"Power of Coset Product is Coset of Power",
"Coset Equals Subgroup iff Element in Subgroup"
] |
proofwiki-15204 | Additive Group of Integers is Normal Subgroup of Reals | Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$. | From Additive Group of Integers is Subgroup of Reals, $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.
As the additive group of real numbers is abelian, from Subgroup of Abelian Group is Normal it follows that $\struct {\Z, +} \lhd \struct {\R, +}$.
{{qed}} | Let $\struct {\Z, +}$ be the [[Definition:Additive Group of Integers|additive group of integers]].
Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Then $\struct {\Z, +}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {\R, +}$. | From [[Additive Group of Integers is Subgroup of Reals]], $\struct {\Z, +}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {\R, +}$.
As the [[Definition:Additive Group of Real Numbers|additive group of real numbers]] is [[Definition:Abelian Group|abelian]], from [[Subgroup of Abelian Group is Normal]... | Additive Group of Integers is Normal Subgroup of Reals | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Normal_Subgroup_of_Reals | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Normal_Subgroup_of_Reals | [
"Additive Group of Integers",
"Additive Group of Real Numbers",
"Examples of Normal Subgroups"
] | [
"Definition:Additive Group of Integers",
"Definition:Additive Group of Real Numbers",
"Definition:Normal Subgroup"
] | [
"Additive Group of Integers is Subgroup of Reals",
"Definition:Normal Subgroup",
"Definition:Additive Group of Real Numbers",
"Definition:Abelian Group",
"Subgroup of Abelian Group is Normal"
] |
proofwiki-15205 | Condition for Element of Quotient Group of Additive Group of Reals by Integers to be of Finite Order | Let $\struct {\R, +}$ be the additive group of real numbers.
Let $\struct {\Z, +}$ be the additive group of integers.
Let $\R / \Z$ denote the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$.
Let $x + \Z$ denote the coset of $\Z$ by $x \in \R$.
Then $x + \Z$ is of finite order {{iff}} $x$ is rational. | From Additive Group of Integers is Normal Subgroup of Reals, we have that $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.
Hence $\R / \Z$ is indeed a quotient group.
By definition of rational number, what is to be proved is:
:$x + \Z$ is of finite order {{iff}}:
:$x = \dfrac m n$
for some $m \in \Z, n \in ... | Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\struct {\Z, +}$ be the [[Definition:Additive Group of Integers|additive group of integers]].
Let $\R / \Z$ denote the [[Definition:Quotient Group|quotient group]] of $\struct {\R, +}$ by $\struct {\Z, +}$.... | From [[Additive Group of Integers is Normal Subgroup of Reals]], we have that $\struct {\Z, +}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {\R, +}$.
Hence $\R / \Z$ is indeed a [[Definition:Quotient Group|quotient group]].
By definition of [[Definition:Rational Number|rational number]], what is ... | Condition for Element of Quotient Group of Additive Group of Reals by Integers to be of Finite Order | https://proofwiki.org/wiki/Condition_for_Element_of_Quotient_Group_of_Additive_Group_of_Reals_by_Integers_to_be_of_Finite_Order | https://proofwiki.org/wiki/Condition_for_Element_of_Quotient_Group_of_Additive_Group_of_Reals_by_Integers_to_be_of_Finite_Order | [
"Integers",
"Real Numbers",
"Examples of Quotient Groups"
] | [
"Definition:Additive Group of Real Numbers",
"Definition:Additive Group of Integers",
"Definition:Quotient Group",
"Definition:Coset",
"Definition:Order of Group Element/Finite",
"Definition:Rational Number"
] | [
"Additive Group of Integers is Normal Subgroup of Reals",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Rational Number",
"Definition:Order of Group Element/Finite",
"Definition:Order of Group Element/Finite",
"Condition for Power of Element of Quotient Group to be Identity"
] |
proofwiki-15206 | Additive Group of Integers is Normal Subgroup of Rationals | Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\Q, +}$ be the additive group of rational numbers.
Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$. | From Additive Group of Integers is Subgroup of Rationals, $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.
From Rational Numbers under Addition form Infinite Abelian Group, $\struct {\Q, +}$ is an abelian group.
From Subgroup of Abelian Group is Normal it follows that $\struct {\Z, +}$ is a normal subgroup of $\st... | Let $\struct {\Z, +}$ be the [[Definition:Additive Group of Integers|additive group of integers]].
Let $\struct {\Q, +}$ be the [[Definition:Additive Group of Rational Numbers|additive group of rational numbers]].
Then $\struct {\Z, +}$ is a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {\Q, +}$. | From [[Additive Group of Integers is Subgroup of Rationals]], $\struct {\Z, +}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\Q, +}$.
From [[Rational Numbers under Addition form Infinite Abelian Group]], $\struct {\Q, +}$ is an [[Definition:Abelian Group|abelian group]].
From [[Subgroup of Abelian Group is Norm... | Additive Group of Integers is Normal Subgroup of Rationals | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Normal_Subgroup_of_Rationals | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Normal_Subgroup_of_Rationals | [
"Additive Group of Integers",
"Additive Group of Rational Numbers",
"Examples of Normal Subgroups"
] | [
"Definition:Additive Group of Integers",
"Definition:Additive Group of Rational Numbers",
"Definition:Normal Subgroup"
] | [
"Additive Group of Integers is Subgroup of Rationals",
"Definition:Subgroup",
"Rational Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Subgroup of Abelian Group is Normal",
"Definition:Normal Subgroup",
"Category:Additive Group of Integers",
"Category:Additive Group ... |
proofwiki-15207 | Mapping from Additive Group of Integers to Powers of Group Element is Homomorphism | Let $\struct {G, \circ}$ be a group.
Let $g \in G$.
Let $\struct {\Z, +}$ denote the additive group of integers.
Let $\phi_g: \struct {\Z, +} \to \struct {G, \circ}$ be the mapping defined as:
:$\forall k \in \Z: \map {\phi_g} k = g^k$
Then $\phi_g$ is a (group) homomorphism. | Let $k, l \in \Z$.
{{begin-eqn}}
{{eqn | l = \map {\phi_g} {k + l}
| r = a^{k + l}
| c =
}}
{{eqn | r = a^k a^l
| c =
}}
{{eqn | r = \map {\phi_g} k \circ \map {\phi_g} l
| c =
}}
{{end-eqn}}
thus proving that $\phi_g$ is a homomorphism as required.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $g \in G$.
Let $\struct {\Z, +}$ denote the [[Definition:Additive Group of Integers|additive group of integers]].
Let $\phi_g: \struct {\Z, +} \to \struct {G, \circ}$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall k \in \Z: \map {\phi_g} k =... | Let $k, l \in \Z$.
{{begin-eqn}}
{{eqn | l = \map {\phi_g} {k + l}
| r = a^{k + l}
| c =
}}
{{eqn | r = a^k a^l
| c =
}}
{{eqn | r = \map {\phi_g} k \circ \map {\phi_g} l
| c =
}}
{{end-eqn}}
thus proving that $\phi_g$ is a [[Definition:Group Homomorphism|homomorphism]] as required.
{{qed}... | Mapping from Additive Group of Integers to Powers of Group Element is Homomorphism | https://proofwiki.org/wiki/Mapping_from_Additive_Group_of_Integers_to_Powers_of_Group_Element_is_Homomorphism | https://proofwiki.org/wiki/Mapping_from_Additive_Group_of_Integers_to_Powers_of_Group_Element_is_Homomorphism | [
"Additive Group of Integers",
"Examples of Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Additive Group of Integers",
"Definition:Mapping",
"Definition:Group Homomorphism"
] | [
"Definition:Group Homomorphism"
] |
proofwiki-15208 | Inner Automorphisms form Subgroup of Symmetric Group | Let $G$ be a group.
Let $\struct {\map \Gamma G, \circ}$ be the symmetric group on $G$.
Let $\Inn G$ denote the inner automorphism group of $G$.
Then:
:$\Inn G \le \struct {\map \Gamma G, \circ}$
where $\le$ denotes the relation of being a subgroup. | An inner automorphism is a permutation on $G$ by definition.
From Inner Automorphisms form Subgroup of Automorphism Group:
:$\Inn G \le \Aut G$
where $\Aut G$ denotes the set of automorphisms of $G$.
From Automorphism Group is Subgroup of Symmetric Group:
:$\Aut G \le \struct {\map \Gamma G, \circ}$
Thus $\Inn G \le \s... | Let $G$ be a [[Definition:Group|group]].
Let $\struct {\map \Gamma G, \circ}$ be the [[Definition:Symmetric Group|symmetric group]] on $G$.
Let $\Inn G$ denote the [[Definition:Inner Automorphism Group|inner automorphism group]] of $G$.
Then:
:$\Inn G \le \struct {\map \Gamma G, \circ}$
where $\le$ denotes the rela... | An [[Definition:Inner Automorphism|inner automorphism]] is a [[Definition:Permutation|permutation]] on $G$ by definition.
From [[Inner Automorphisms form Subgroup of Automorphism Group]]:
:$\Inn G \le \Aut G$
where $\Aut G$ denotes the set of [[Definition:Group Automorphism|automorphisms]] of $G$.
From [[Automorphism... | Inner Automorphisms form Subgroup of Symmetric Group | https://proofwiki.org/wiki/Inner_Automorphisms_form_Subgroup_of_Symmetric_Group | https://proofwiki.org/wiki/Inner_Automorphisms_form_Subgroup_of_Symmetric_Group | [
"Inner Automorphisms",
"Symmetric Groups"
] | [
"Definition:Group",
"Definition:Symmetric Group",
"Definition:Inner Automorphism Group",
"Definition:Subgroup"
] | [
"Definition:Inner Automorphism",
"Definition:Permutation",
"Inner Automorphisms form Subgroup of Automorphism Group",
"Definition:Group Automorphism",
"Automorphism Group is Subgroup of Symmetric Group"
] |
proofwiki-15209 | Group Isomorphism Preserves Order of Group | Let $G$ and $H$ be groups.
Let $\phi: G \to H$ be a (group) isomorphism.
Then:
:$\order G = \order H$
where $\order {\, \cdot \,}$ denotes the order of a group. | By definition, an isomorphism is a bijection.
By definition, the order of a group is the cardinality of its underlying set.
The result follows by definition of set equivalence.
{{qed}} | Let $G$ and $H$ be [[Definition:Group|groups]].
Let $\phi: G \to H$ be a [[Definition:Group Isomorphism|(group) isomorphism]].
Then:
:$\order G = \order H$
where $\order {\, \cdot \,}$ denotes the [[Definition:Order of Group|order]] of a [[Definition:Group|group]]. | By definition, an [[Definition:Group Isomorphism|isomorphism]] is a [[Definition:Bijection|bijection]].
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]].
The result foll... | Group Isomorphism Preserves Order of Group | https://proofwiki.org/wiki/Group_Isomorphism_Preserves_Order_of_Group | https://proofwiki.org/wiki/Group_Isomorphism_Preserves_Order_of_Group | [
"Group Isomorphisms"
] | [
"Definition:Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Order of Structure",
"Definition:Group"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Bijection",
"Definition:Order of Structure",
"Definition:Group",
"Definition:Cardinality",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Set Equivalence"
] |
proofwiki-15210 | Additive Group of Reals is Subgroup of Complex | Let $\struct {\R, +}$ be the additive group of real numbers.
Let $\struct {\C, +}$ be the additive group of complex numbers.
Then $\struct {\R, +}$ is a subgroup of $\struct {\C, +}$. | Let $x, y \in \C$ such that $x = x_1 + 0 i, y = y_1 + 0 i$.
As $x$ and $y$ are wholly real, we have that $x, y \in \R$.
Then $x + y = \paren {x_1 + y_1} + \paren {0 + 0} i$ which is also wholly real.
Also, the inverse of $x$ is $-x = -x_1 + 0 i$ which is also wholly real.
We have that $\R$ is non-empty.
Thus by the Two... | Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\struct {\C, +}$ be the [[Definition:Additive Group of Complex Numbers|additive group of complex numbers]].
Then $\struct {\R, +}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\C, +}$. | Let $x, y \in \C$ such that $x = x_1 + 0 i, y = y_1 + 0 i$.
As $x$ and $y$ are [[Definition:Wholly Real|wholly real]], we have that $x, y \in \R$.
Then $x + y = \paren {x_1 + y_1} + \paren {0 + 0} i$ which is also [[Definition:Wholly Real|wholly real]].
Also, the inverse of $x$ is $-x = -x_1 + 0 i$ which is also [[D... | Additive Group of Reals is Subgroup of Complex | https://proofwiki.org/wiki/Additive_Group_of_Reals_is_Subgroup_of_Complex | https://proofwiki.org/wiki/Additive_Group_of_Reals_is_Subgroup_of_Complex | [
"Additive Group of Real Numbers",
"Additive Group of Complex Numbers"
] | [
"Definition:Additive Group of Real Numbers",
"Definition:Additive Group of Complex Numbers",
"Definition:Subgroup"
] | [
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Real",
"Definition:Non-Empty Set",
"Two-Step Subgroup Test",
"Definition:Subgroup"
] |
proofwiki-15211 | C6 is not Isomorphic to S3 | Let $C_6$ denote the cyclic group of order $6$.
Let $S_3$ denote the symmetric group on $3$ letters.
Then $C_6$ and $S_3$ are not isomorphic. | Note that both $C_6$ and $S_3$ are of order $6$.
From Cyclic Group is Abelian, $C_6$ is abelian.
From Symmetric Group is not Abelian, $S_6$ is not abelian.
From Isomorphism of Abelian Groups, if two groups are isomorphic, they are either both abelian or both not abelian.
Hence $C_6$ and $S_3$ are not isomorphic.
{{qed}... | Let $C_6$ denote the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order $6$]].
Let $S_3$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $3$ letters]].
Then $C_6$ and $S_3$ are not [[Definition:Group Isomorphism|isomorphic]]. | Note that both $C_6$ and $S_3$ are of [[Definition:Order of Group|order $6$]].
From [[Cyclic Group is Abelian]], $C_6$ is [[Definition:Abelian Group|abelian]].
From [[Symmetric Group is not Abelian]], $S_6$ is not [[Definition:Abelian Group|abelian]].
From [[Isomorphism of Abelian Groups]], if two [[Definition:Group... | C6 is not Isomorphic to S3 | https://proofwiki.org/wiki/C6_is_not_Isomorphic_to_S3 | https://proofwiki.org/wiki/C6_is_not_Isomorphic_to_S3 | [
"Examples of Cyclic Groups",
"Symmetric Group on 3 Letters"
] | [
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Symmetric Group/n Letters",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Order of Structure",
"Cyclic Group is Abelian",
"Definition:Abelian Group",
"Symmetric Group is not Abelian",
"Definition:Abelian Group",
"Isomorphism of Abelian Groups",
"Definition:Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Abelian Group",
"Def... |
proofwiki-15212 | Order of Alternating Group | Let $n \in \Z$ be an integer such that $n > 1$.
Let $A_n$ be the alternating group on $n$ letters.
Then:
:$\order {A_n} = \dfrac {n!} 2$
where $\order {A_n}$ denotes the order of $A_n$. | Let $S_n$ denote the symmetric group on $n$ letters.
From Alternating Group is Normal Subgroup of Symmetric Group:
:$\index {S_n} {A_n} = 2$
where $\index {S_n} {A_n}$ denotes the index of $A_n$ in $S_n$.
From Order of Symmetric Group:
:$\order {S_n} = n!$
The result follows from Lagrange's Theorem.
{{qed}} | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n > 1$.
Let $A_n$ be the [[Definition:Alternating Group|alternating group on $n$ letters]].
Then:
:$\order {A_n} = \dfrac {n!} 2$
where $\order {A_n}$ denotes the [[Definition:Order of Group|order]] of $A_n$. | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
From [[Alternating Group is Normal Subgroup of Symmetric Group]]:
:$\index {S_n} {A_n} = 2$
where $\index {S_n} {A_n}$ denotes the [[Definition:Index of Subgroup|index]] of $A_n$ in $S_n$.
From [[Order of Symmetric Group]... | Order of Alternating Group | https://proofwiki.org/wiki/Order_of_Alternating_Group | https://proofwiki.org/wiki/Order_of_Alternating_Group | [
"Alternating Groups"
] | [
"Definition:Integer",
"Definition:Alternating Group",
"Definition:Order of Structure"
] | [
"Definition:Symmetric Group/n Letters",
"Alternating Group is Normal Subgroup of Symmetric Group",
"Definition:Index of Subgroup",
"Order of Symmetric Group",
"Lagrange's Theorem (Group Theory)"
] |
proofwiki-15213 | Mapping to Power is Endomorphism iff Abelian | Let $\struct {G, \circ}$ be a group.
Let $n \in \Z$ be an integer.
Let $\phi: G \to G$ be defined as:
:$\forall g \in G: \map \phi g = g^n$
Then $\struct {G, \circ}$ is abelian {{iff}} $\phi$ is a (group) endomorphism. | === Necessary Condition ===
Let $\struct {G, \circ}$ be an abelian group.
Let $a, b \in G$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi {a \circ b}
| r = \paren {a \circ b}^n
| c = Definition of $\phi$
}}
{{eqn | r = a^n \circ b^n
| c = Power of Product of Commutative Elements in Group
}}
{... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $n \in \Z$ be an [[Definition:Integer|integer]].
Let $\phi: G \to G$ be defined as:
:$\forall g \in G: \map \phi g = g^n$
Then $\struct {G, \circ}$ is [[Definition:Abelian Group|abelian]] {{iff}} $\phi$ is a [[Definition:Group Endomorphism|(group) endomo... | === Necessary Condition ===
Let $\struct {G, \circ}$ be an [[Definition:Abelian Group|abelian group]].
Let $a, b \in G$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi {a \circ b}
| r = \paren {a \circ b}^n
| c = Definition of $\phi$
}}
{{eqn | r = a^n \circ b^n
| c = [[Power of Product of ... | Mapping to Power is Endomorphism iff Abelian | https://proofwiki.org/wiki/Mapping_to_Power_is_Endomorphism_iff_Abelian | https://proofwiki.org/wiki/Mapping_to_Power_is_Endomorphism_iff_Abelian | [
"Abelian Groups",
"Group Endomorphisms"
] | [
"Definition:Group",
"Definition:Integer",
"Definition:Abelian Group",
"Definition:Group Endomorphism"
] | [
"Definition:Abelian Group",
"Power of Product of Commutative Elements in Group",
"Definition:Group Homomorphism",
"Definition:Group Endomorphism",
"Definition:Group Endomorphism",
"Power of Product of Commutative Elements in Group",
"Definition:Abelian Group"
] |
proofwiki-15214 | Additive Groups of Integers and Integer Multiples are Isomorphic | Let $n \in \Z_{> 0}$ be a strictly positive integer.
Let $\struct {n \Z, +}$ denote the additive group of integer multiples.
Let $\struct {\Z, +}$ denote the additive group of integers.
Then $\struct {n \Z, +}$ is isomorphic to $\struct {\Z, +}$. | We have that:
:Infinite Cyclic Group is Isomorphic to Integers.
:Integer Multiples under Addition form Infinite Cyclic Group.
:Infinite Cyclic Group is Unique up to Isomorphism
Hence the result.
{{qed}} | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\struct {n \Z, +}$ denote the [[Definition:Additive Group of Integer Multiples|additive group of integer multiples]].
Let $\struct {\Z, +}$ denote the [[Definition:Additive Group of Integers|additive group of integers]]... | We have that:
:[[Infinite Cyclic Group is Isomorphic to Integers]].
:[[Integer Multiples under Addition form Infinite Cyclic Group]].
:[[Infinite Cyclic Group is Unique up to Isomorphism]]
Hence the result.
{{qed}} | Additive Groups of Integers and Integer Multiples are Isomorphic | https://proofwiki.org/wiki/Additive_Groups_of_Integers_and_Integer_Multiples_are_Isomorphic | https://proofwiki.org/wiki/Additive_Groups_of_Integers_and_Integer_Multiples_are_Isomorphic | [
"Additive Groups of Integer Multiples",
"Additive Group of Integers",
"Infinite Cyclic Group"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Additive Group of Integer Multiples",
"Definition:Additive Group of Integers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Infinite Cyclic Group is Isomorphic to Integers",
"Integer Multiples under Addition form Infinite Cyclic Group",
"Infinite Cyclic Group is Unique up to Isomorphism"
] |
proofwiki-15215 | Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers | Let $\struct {\R, +}$ denote the additive group of real numbers.
Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers.
Then $\struct {\R, +}$ is not isomorphic to $\struct {\R_{\ne 0}, \times}$. | Consider the element $-1 \in \struct {\R_{\ne 0}, \times}$.
We have that:
:$-1 \times -1 = 1$
From Real Multiplication Identity is One it follows that $-1$ is of order $2$ in $\struct {\R_{\ne 0}, \times}$.
From Group Isomorphism Preserves Order of Group Element, it is sufficient to demonstrate that there exists no ele... | Let $\struct {\R, +}$ denote the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\struct {\R_{\ne 0}, \times}$ denote the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of real numbers]].
Then $\struct {\R, +}$ is not [[Definition:Group Isomorphism|isomorphi... | Consider the [[Definition:Element|element]] $-1 \in \struct {\R_{\ne 0}, \times}$.
We have that:
:$-1 \times -1 = 1$
From [[Real Multiplication Identity is One]] it follows that $-1$ is of [[Definition:Order of Group Element|order $2$]] in $\struct {\R_{\ne 0}, \times}$.
From [[Group Isomorphism Preserves Order of ... | Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers/Proof 1 | https://proofwiki.org/wiki/Additive_Group_of_Real_Numbers_is_Not_Isomorphic_to_Multiplicative_Group_of_Real_Numbers | https://proofwiki.org/wiki/Additive_Group_of_Real_Numbers_is_Not_Isomorphic_to_Multiplicative_Group_of_Real_Numbers/Proof_1 | [
"Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers",
"Multiplicative Group of Real Numbers",
"Additive Group of Real Numbers",
"Examples of Isomorphisms (Abstract Algebra)"
] | [
"Definition:Additive Group of Real Numbers",
"Definition:Multiplicative Group of Real Numbers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Element",
"Real Multiplication Identity is One",
"Definition:Order of Group Element",
"Group Isomorphism Preserves Order of Group Element",
"Definition:Element",
"Definition:Order of Group Element",
"Real Addition Identity is Zero",
"Definition:Identity (Abstract Algebra)/Two-Sided Identit... |
proofwiki-15216 | Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers | Let $\struct {\R, +}$ denote the additive group of real numbers.
Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers.
Then $\struct {\R, +}$ is not isomorphic to $\struct {\R_{\ne 0}, \times}$. | There are two element of $\struct {\R_{\ne 0}, \times}$ which are self-inverse, that is, $-1$ and $1$.
However, there is only one element of $\struct {\R, +}$ which is self-inverse, that is, $0$.
{{AimForCont}} there exists an isomorphism $f: \struct {\R_{\ne 0}, \times} \to \struct {\R, +}$.
Then:
{{begin-eqn}}
{{eqn ... | Let $\struct {\R, +}$ denote the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\struct {\R_{\ne 0}, \times}$ denote the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of real numbers]].
Then $\struct {\R, +}$ is not [[Definition:Group Isomorphism|isomorphi... | There are two [[Definition:Element|element]] of $\struct {\R_{\ne 0}, \times}$ which are [[Definition:Self-Inverse Element|self-inverse]], that is, $-1$ and $1$.
However, there is only one [[Definition:Element|element]] of $\struct {\R, +}$ which is [[Definition:Self-Inverse Element|self-inverse]], that is, $0$.
{{Ai... | Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers/Proof 3 | https://proofwiki.org/wiki/Additive_Group_of_Real_Numbers_is_Not_Isomorphic_to_Multiplicative_Group_of_Real_Numbers | https://proofwiki.org/wiki/Additive_Group_of_Real_Numbers_is_Not_Isomorphic_to_Multiplicative_Group_of_Real_Numbers/Proof_3 | [
"Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers",
"Multiplicative Group of Real Numbers",
"Additive Group of Real Numbers",
"Examples of Isomorphisms (Abstract Algebra)"
] | [
"Definition:Additive Group of Real Numbers",
"Definition:Multiplicative Group of Real Numbers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Element",
"Definition:Self-Inverse Element",
"Definition:Element",
"Definition:Self-Inverse Element",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Injection",
"Definition:Bijection",
"Definition:Isomorphism (Abstract Algebra)",
"Proof by Contradiction",
"Definition:Isomorph... |
proofwiki-15217 | Normal Subgroup is Kernel of Group Homomorphism | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Then there exists a group homomorphism of which $N$ is the kernel. | Let $G / N$ be the quotient group of $G$ by $N$.
Let $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$:
:$\forall x \in G: \map {q_N} x = x N$
Then from Quotient Group Epimorphism is Epimorphism, $N$ is the kernel of $q_n$
Thus $q_N$ is that group homomorphism of which $N$ is the kernel.
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Then there exists a [[Definition:Group Homomorphism|group homomorphism]] of which $N$ is the [[Definition:Kernel of Group Homomorphism|kernel]]. | Let $G / N$ be the [[Definition:Quotient Group|quotient group]] of $G$ by $N$.
Let $q_N: G \to G / N$ be the [[Definition:Quotient Group Epimorphism|quotient epimorphism]] from $G$ to $G / N$:
:$\forall x \in G: \map {q_N} x = x N$
Then from [[Quotient Group Epimorphism is Epimorphism]], $N$ is the [[Definition:Kerne... | Normal Subgroup is Kernel of Group Homomorphism | https://proofwiki.org/wiki/Normal_Subgroup_is_Kernel_of_Group_Homomorphism | https://proofwiki.org/wiki/Normal_Subgroup_is_Kernel_of_Group_Homomorphism | [
"Normal Subgroups",
"Group Homomorphisms",
"Kernels of Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism"
] | [
"Definition:Quotient Group",
"Definition:Quotient Epimorphism/Group",
"Quotient Epimorphism is Epimorphism/Group",
"Definition:Kernel of Group Homomorphism",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism"
] |
proofwiki-15218 | Homomorphic Image of Cyclic Group is Cyclic Group | Let $G$ be a cyclic group with generator $g$.
Let $H$ be a group.
Let $\phi: G \to H$ be a (group) homomorphism.
Let $\Img G$ denote the homomorphic image of $G$ under $\phi$.
Then $\Img G$ is a cyclic group with generator $\map \phi g$.
That is:
:$\phi \sqbrk {\gen g} = \gen {\map \phi g}$ | Let $y \in \Img G$.
Then $\exists x \in G: y = \map \phi x$.
As $G$ be a cyclic group with generator $g$, $x = g^n$ for some $n \in \Z$.
Thus by Homomorphism of Power of Group Element:
:$y = \paren {\map \phi g}^n$
and so is a power of $\map \phi g$.
As $y$ is arbitrary, it follows that all elements of $\Img G$ are pow... | Let $G$ be a [[Definition:Cyclic Group|cyclic group]] with [[Definition:Generator of Cyclic Group|generator]] $g$.
Let $H$ be a [[Definition:Group|group]].
Let $\phi: G \to H$ be a [[Definition:Group Homomorphism|(group) homomorphism]].
Let $\Img G$ denote the [[Definition:Homomorphic Image|homomorphic image]] of $G... | Let $y \in \Img G$.
Then $\exists x \in G: y = \map \phi x$.
As $G$ be a [[Definition:Cyclic Group|cyclic group]] with [[Definition:Generator of Cyclic Group|generator]] $g$, $x = g^n$ for some $n \in \Z$.
Thus by [[Homomorphism of Power of Group Element]]:
:$y = \paren {\map \phi g}^n$
and so is a [[Definition:Pow... | Homomorphic Image of Cyclic Group is Cyclic Group | https://proofwiki.org/wiki/Homomorphic_Image_of_Cyclic_Group_is_Cyclic_Group | https://proofwiki.org/wiki/Homomorphic_Image_of_Cyclic_Group_is_Cyclic_Group | [
"Cyclic Groups",
"Group Homomorphisms"
] | [
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator",
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Homomorphism (Abstract Algebra)/Image",
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator"
] | [
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator",
"Homomorphism of Power of Group Element",
"Definition:Power of Element/Group",
"Definition:Element",
"Definition:Power of Element/Group",
"Definition:Cyclic Group"
] |
proofwiki-15219 | Power of Group Element in Kernel of Homomorphism iff Power of Image is Identity | Let $G$ be a group whose identity is $e_G$.
Let $H$ be a group whose identity is $e_H$.
Let $\phi: G \to H$ be a (group) homomorphism.
Let $x^n \in \map \ker \phi$ for some integer $n$.
Then:
:$\paren {\map \phi x}^n = e_H$ | {{begin-eqn}}
{{eqn | l = x^n
| o = \in
| r = \map \ker \phi
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map \phi {x^n}
| r = e_H
| c = {{Defof|Kernel of Group Homomorphism}}
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {\map \phi x}^n
| r = e_H
| c = Homomorphism o... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e_G$.
Let $H$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e_H$.
Let $\phi: G \to H$ be a [[Definition:Group Homomorphism|(group) homomorphism]].
Let $x^n \in \map \ker \phi$ for some [[De... | {{begin-eqn}}
{{eqn | l = x^n
| o = \in
| r = \map \ker \phi
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map \phi {x^n}
| r = e_H
| c = {{Defof|Kernel of Group Homomorphism}}
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {\map \phi x}^n
| r = e_H
| c = [[Homomorphism... | Power of Group Element in Kernel of Homomorphism iff Power of Image is Identity | https://proofwiki.org/wiki/Power_of_Group_Element_in_Kernel_of_Homomorphism_iff_Power_of_Image_is_Identity | https://proofwiki.org/wiki/Power_of_Group_Element_in_Kernel_of_Homomorphism_iff_Power_of_Image_is_Identity | [
"Kernels of Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Homomorphism",
"Definition:Integer"
] | [
"Homomorphism of Power of Group Element"
] |
proofwiki-15220 | Kernel of Homomorphism on Cyclic Group | Let $G = \gen g$ be a cyclic group with generator $g$.
Let $H$ be a group.
Let $\phi: G \to H$ be a (group) homomorphism.
Let $\map \ker \phi$ denote the kernel of $\phi$.
Let $\Img G$ denote the homomorphic image of $G$ under $\phi$.
Then:
:$\map \ker \phi = \gen {g^m}$
where:
:$m = 0$ if $\Img \phi$ is an infinite cy... | From Kernel of Group Homomorphism is Subgroup and Subgroup of Cyclic Group is Cyclic:
:$\exists m \in \N: \map \ker \phi = \gen {g^m}$
From Homomorphic Image of Cyclic Group is Cyclic Group:
:$\Img \phi$ is a cyclic group generated by $\map \phi g$. | Let $G = \gen g$ be a [[Definition:Cyclic Group|cyclic group]] with [[Definition:Generator of Cyclic Group|generator]] $g$.
Let $H$ be a [[Definition:Group|group]].
Let $\phi: G \to H$ be a [[Definition:Group Homomorphism|(group) homomorphism]].
Let $\map \ker \phi$ denote the [[Definition:Kernel of Group Homomorphi... | From [[Kernel of Group Homomorphism is Subgroup]] and [[Subgroup of Cyclic Group is Cyclic]]:
:$\exists m \in \N: \map \ker \phi = \gen {g^m}$
From [[Homomorphic Image of Cyclic Group is Cyclic Group]]:
:$\Img \phi$ is a [[Definition:Cyclic Group|cyclic group]] [[Definition:Generator of Cyclic Group|generated by]] $\m... | Kernel of Homomorphism on Cyclic Group | https://proofwiki.org/wiki/Kernel_of_Homomorphism_on_Cyclic_Group | https://proofwiki.org/wiki/Kernel_of_Homomorphism_on_Cyclic_Group | [
"Cyclic Groups",
"Kernels of Group Homomorphisms"
] | [
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator",
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Homomorphism (Abstract Algebra)/Image",
"Definition:Infinite Cyclic Group",
"Definition:Finite Cyclic Group"
] | [
"Kernel of Group Homomorphism is Subgroup",
"Subgroup of Cyclic Group is Cyclic",
"Homomorphic Image of Cyclic Group is Cyclic Group",
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator"
] |
proofwiki-15221 | Mapping from Group Element to Inner Automorphism is Homomorphism | Let $G$ be a group.
Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as:
:$\forall x \in G: \map \kappa x := \kappa_x$
where $\kappa_x$ is the inner automorphism on $x$:
:$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
Then $\kappa$ is a homomorphism. | Let $x, y \in G$.
By definition of automorphism group, we have that:
:$\map \kappa x \map \kappa y = \kappa_x \circ \kappa_y$
where $\circ$ denotes composition of mappings.
Then $\forall g \in G$:
{{begin-eqn}}
{{eqn | l = \map {\kappa_x \circ \kappa_y} g
| r = \map {\kappa_x} {\map {\kappa_y} g}
| c = for ... | Let $G$ be a [[Definition:Group|group]].
Let $\kappa: G \to \Aut G$ be the [[Definition:Mapping|mapping]] from $G$ to the [[Definition:Automorphism Group of Group|automorphism group]] of $G$ defined as:
:$\forall x \in G: \map \kappa x := \kappa_x$
where $\kappa_x$ is the [[Definition:Inner Automorphism|inner automor... | Let $x, y \in G$.
By definition of [[Definition:Automorphism Group of Group|automorphism group]], we have that:
:$\map \kappa x \map \kappa y = \kappa_x \circ \kappa_y$
where $\circ$ denotes [[Definition:Composition of Mappings|composition of mappings]].
Then $\forall g \in G$:
{{begin-eqn}}
{{eqn | l = \map {\kapp... | Mapping from Group Element to Inner Automorphism is Homomorphism | https://proofwiki.org/wiki/Mapping_from_Group_Element_to_Inner_Automorphism_is_Homomorphism | https://proofwiki.org/wiki/Mapping_from_Group_Element_to_Inner_Automorphism_is_Homomorphism | [
"Inner Automorphisms",
"Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Mapping",
"Definition:Automorphism Group/Group",
"Definition:Inner Automorphism",
"Definition:Group Homomorphism"
] | [
"Definition:Automorphism Group/Group",
"Definition:Composition of Mappings",
"Inverse of Group Product",
"Definition:Group Homomorphism"
] |
proofwiki-15222 | Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group | Let $G$ be a group.
Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as:
:$\forall x \in G: \map \kappa x := \kappa_x$
where $\kappa_x$ is the inner automorphism on $x$:
:$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
Then $\Img \kappa$ is the inner automorphism group of $G... | Let $\Inn G$ denote the inner automorphism group of $G$.
For all $x \in G$, $\map \kappa x = \kappa_x \in \Inn G$.
Hence $\Img \kappa \subseteq \Inn G$.
Let $\phi \in \Inn G$. Then:
:$\exists y \in G: \forall g \in G: \map \phi g = y g y^{-1}$
Then $\map \kappa y = \phi$.
Hence $\Inn G \subseteq \Img \kappa$.
Therefo... | Let $G$ be a [[Definition:Group|group]].
Let $\kappa: G \to \Aut G$ be the [[Definition:Mapping|mapping]] from $G$ to the [[Definition:Automorphism Group of Group|automorphism group]] of $G$ defined as:
:$\forall x \in G: \map \kappa x := \kappa_x$
where $\kappa_x$ is the [[Definition:Inner Automorphism|inner automor... | Let $\Inn G$ denote the [[Definition:Inner Automorphism Group|inner automorphism group]] of $G$.
For all $x \in G$, $\map \kappa x = \kappa_x \in \Inn G$.
Hence $\Img \kappa \subseteq \Inn G$.
Let $\phi \in \Inn G$. Then:
:$\exists y \in G: \forall g \in G: \map \phi g = y g y^{-1}$
Then $\map \kappa y = \phi$.... | Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group | https://proofwiki.org/wiki/Image_of_Mapping_from_Group_Element_to_Inner_Automorphism_is_Inner_Automorphism_Group | https://proofwiki.org/wiki/Image_of_Mapping_from_Group_Element_to_Inner_Automorphism_is_Inner_Automorphism_Group | [
"Inner Automorphisms",
"Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Mapping",
"Definition:Automorphism Group/Group",
"Definition:Inner Automorphism",
"Definition:Inner Automorphism Group"
] | [
"Definition:Inner Automorphism Group"
] |
proofwiki-15223 | Order of Monomorphic Image of Group Element | Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a monomorphism.
Let $g \in G$ be of finite order.
Then:
:$\forall g \in G: \order {\map \phi g} = \order g$ | By definition of monomorphism, $\phi$ is a homomorphism which is also an injection.
From Order of Homomorphic Image of Group Element:
:$\forall g \in G: \order {\map \phi g} \divides \order g$
{{begin-eqn}}
{{eqn | l = \map \phi {g^m}
| r = \paren {\map \phi g}^m
| c = Homomorphism of Power of Group Element... | Let $G$ and $H$ be [[Definition:Group|groups]] whose [[Definition:Identity Element|identities]] are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a [[Definition:Group Monomorphism|monomorphism]].
Let $g \in G$ be of [[Definition:Order of Group Element|finite order]].
Then:
:$\forall g \in G: \order {\map \ph... | By definition of [[Definition:Group Monomorphism|monomorphism]], $\phi$ is a [[Definition:Group Homomorphism|homomorphism]] which is also an [[Definition:Injection|injection]].
From [[Order of Homomorphic Image of Group Element]]:
:$\forall g \in G: \order {\map \phi g} \divides \order g$
{{begin-eqn}}
{{eqn | l = \... | Order of Monomorphic Image of Group Element | https://proofwiki.org/wiki/Order_of_Monomorphic_Image_of_Group_Element | https://proofwiki.org/wiki/Order_of_Monomorphic_Image_of_Group_Element | [
"Group Monomorphisms"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Monomorphism",
"Definition:Order of Group Element"
] | [
"Definition:Group Monomorphism",
"Definition:Group Homomorphism",
"Definition:Injection",
"Order of Homomorphic Image of Group Element",
"Homomorphism of Power of Group Element",
"Homomorphism to Group Preserves Identity",
"Definition:Injection",
"Definition:Order of Group Element"
] |
proofwiki-15224 | Image under Epimorphism of Center is Subset of Center | Let $G$ and $H$ be groups.
Let $\theta: G \to H$ be an epimorphism.
Let $\map Z G$ denote the center of $G$.
Then:
:$\theta \sqbrk {\map Z G} \subseteq \map Z H$ | Let $y \in \theta \sqbrk {\map Z G}$.
Let $t \in H$.
We have that:
:$y = \map \theta z$
for some $z \in \map Z G$
As $\theta$ is an epimorphism, it is by definition surjective.
Then:
:$t = \map \theta s$
for some $s \in G$.
Hence:
{{begin-eqn}}
{{eqn | l = y t
| r = \map \theta z \map \theta s
| c =
}}
{{e... | Let $G$ and $H$ be [[Definition:Group|groups]].
Let $\theta: G \to H$ be an [[Definition:Group Epimorphism|epimorphism]].
Let $\map Z G$ denote the [[Definition:Center of Group|center]] of $G$.
Then:
:$\theta \sqbrk {\map Z G} \subseteq \map Z H$ | Let $y \in \theta \sqbrk {\map Z G}$.
Let $t \in H$.
We have that:
:$y = \map \theta z$
for some $z \in \map Z G$
As $\theta$ is an [[Definition:Group Epimorphism|epimorphism]], it is by definition [[Definition:Surjection|surjective]].
Then:
:$t = \map \theta s$
for some $s \in G$.
Hence:
{{begin-eqn}}
{{eqn | l ... | Image under Epimorphism of Center is Subset of Center | https://proofwiki.org/wiki/Image_under_Epimorphism_of_Center_is_Subset_of_Center | https://proofwiki.org/wiki/Image_under_Epimorphism_of_Center_is_Subset_of_Center | [
"Centers of Groups",
"Group Epimorphisms"
] | [
"Definition:Group",
"Definition:Group Epimorphism",
"Definition:Center (Abstract Algebra)/Group"
] | [
"Definition:Group Epimorphism",
"Definition:Surjection"
] |
proofwiki-15225 | Metric Subspace Induces Subspace Topology | Let $M = \struct {A,d}$ be a metric space.
Let $H \subseteq A$.
Let $\tau$ be the topology induced by the metric $d$.
Let $\tau_H$ be the subspace topology induced by $\tau$ on $H$.
Let $d_H$ be the subspace metric induced by $d$ on $H$.
Let $\tau_{d_H}$ be the topology induced by the metric $d_H$.
Then:
:$\tau_{d_H} =... | Let $\BB$ be the set of open $\epsilon$-balls in $M$.
Let $\BB_H$ be the set of open $\epsilon$-balls in $\struct {H, d_H}$.
Let $U \in \tau_{d_H}$.
By the definition of the topology induced by the metric $d_H$:
:$\exists \AA_H \subseteq \BB_H: U = \bigcup \AA_H$
Let $\AA = \set {B': B' \in \BB, B' \cap H \in \AA_H}$.
... | Let $M = \struct {A,d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$.
Let $\tau$ be the [[Definition:Topology Induced by Metric|topology induced by the metric $d$]].
Let $\tau_H$ be the [[Definition:Subspace Topology|subspace topology]] induced by $\tau$ on $H$.
Let $d_H$ be the [[Definition:M... | Let $\BB$ be the set of [[Definition:Open Ball of Metric Space|open $\epsilon$-balls]] in $M$.
Let $\BB_H$ be the set of [[Definition:Open Ball of Metric Space|open $\epsilon$-balls]] in $\struct {H, d_H}$.
Let $U \in \tau_{d_H}$.
By the definition of the [[Definition:Topology Induced by Metric|topology induced by ... | Metric Subspace Induces Subspace Topology | https://proofwiki.org/wiki/Metric_Subspace_Induces_Subspace_Topology | https://proofwiki.org/wiki/Metric_Subspace_Induces_Subspace_Topology | [
"Metric Subspaces",
"Topological Subspaces"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Topological Subspace",
"Definition:Metric Subspace",
"Definition:Topology Induced by Metric"
] | [
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Topology Induced by Metric",
"Definition:Topology Induced by Metric",
"Definition:Metric Subspace",
"Intersection Distributes over Union",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Topology Induced by... |
proofwiki-15226 | Index of Intersection of Subgroups/Corollary | Let $H$ be a subgroup of $G$.
Let $K$ be a subgroup of finite index of $G$.
Then:
:$\index H {H \cap K} \le \index G K$ | Note that $H \cap K$ is a subgroup of $H$.
From Index of Intersection of Subgroups, we have:
:$\index G {H \cap K} \le \index G H \index G K$
Setting $G = H$, we have:
:$\index H {H \cap K} \le \index H H \index H K$
{{finish|This does not get us where we want}}
{{CircularStructure|This is Index in Subgroup, which is u... | Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $K$ be a [[Definition:Subgroup|subgroup]] of [[Definition:Finite Index|finite index]] of $G$.
Then:
:$\index H {H \cap K} \le \index G K$ | Note that $H \cap K$ is a [[Definition:Subgroup|subgroup]] of $H$.
From [[Index of Intersection of Subgroups]], we have:
:$\index G {H \cap K} \le \index G H \index G K$
Setting $G = H$, we have:
:$\index H {H \cap K} \le \index H H \index H K$
{{finish|This does not get us where we want}}
{{CircularStructure|This... | Index of Intersection of Subgroups/Corollary | https://proofwiki.org/wiki/Index_of_Intersection_of_Subgroups/Corollary | https://proofwiki.org/wiki/Index_of_Intersection_of_Subgroups/Corollary | [
"Index of Intersection of Subgroups"
] | [
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Index of Subgroup/Finite"
] | [
"Definition:Subgroup",
"Index of Intersection of Subgroups",
"Index in Subgroup"
] |
proofwiki-15227 | Quotient of Cauchy Sequences is Metric Completion/Lemma 1 | :$\quad \CC \,\big / \NN = \tilde \CC$ | Let $\sequence {x_n}$ and $\sequence {y_n}$ be Cauchy sequences in $\CC$.
Then:
{{begin-eqn}}
{{eqn | l = \sequence {x_n} + \NN = \sequence {y_n} + \NN
| o = \leadstoandfrom
| r = \sequence {x_n} - \sequence {y_n} \in \NN
| c = Cosets are Equal iff Product with Inverse in Subgroup
}}
{{eqn | r = \lim_... | :$\quad \CC \,\big / \NN = \tilde \CC$ | Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequences]] in $\CC$.
Then:
{{begin-eqn}}
{{eqn | l = \sequence {x_n} + \NN = \sequence {y_n} + \NN
| o = \leadstoandfrom
| r = \sequence {x_n} - \sequence {y_n} \in \NN
| c = [[Cosets are Equa... | Quotient of Cauchy Sequences is Metric Completion/Lemma 1 | https://proofwiki.org/wiki/Quotient_of_Cauchy_Sequences_is_Metric_Completion/Lemma_1 | https://proofwiki.org/wiki/Quotient_of_Cauchy_Sequences_is_Metric_Completion/Lemma_1 | [
"Completion of Normed Division Ring"
] | [] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Cosets are Equal iff Product with Inverse in Subgroup",
"Definition:Equivalence Class",
"Definition:Equivalence Class"
] |
proofwiki-15228 | Quotient of Cauchy Sequences is Metric Completion/Lemma 2 | :$\quad d' = \tilde d$ | By Lemma 1 of Quotient of Cauchy Sequences is Metric Completion we have that:
:$\CC \,\big / \NN = \tilde {\CC}$
Let $\eqclass {x_n} {}$ and $\eqclass {x_n} {}$ be equivalence classes in $\CC \,\big / \NN = \tilde {\CC}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d'} {\eqclass {x_n}{}, \eqclass {x_n}{} }
| r = \norm ... | :$\quad d' = \tilde d$ | By [[Quotient of Cauchy Sequences is Metric Completion/Lemma 1|Lemma 1 of Quotient of Cauchy Sequences is Metric Completion]] we have that:
:$\CC \,\big / \NN = \tilde {\CC}$
Let $\eqclass {x_n} {}$ and $\eqclass {x_n} {}$ be [[Definition:Equivalence Class|equivalence classes]] in $\CC \,\big / \NN = \tilde {\CC}$.
T... | Quotient of Cauchy Sequences is Metric Completion/Lemma 2 | https://proofwiki.org/wiki/Quotient_of_Cauchy_Sequences_is_Metric_Completion/Lemma_2 | https://proofwiki.org/wiki/Quotient_of_Cauchy_Sequences_is_Metric_Completion/Lemma_2 | [
"Completion of Normed Division Ring"
] | [] | [
"Quotient of Cauchy Sequences is Metric Completion/Lemma 1",
"Definition:Equivalence Class",
"Definition:Norm/Division Ring"
] |
proofwiki-15229 | Generator for Quaternion Group | The Quaternion Group can be generated by the matrices:
:$\mathbf a = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf b = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$
where $i$ is the imaginary unit:
:$i^2 = -1$ | Note that:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
is the identity for (conventional) matrix multiplication of order $2$.
We have:
{{begin-eqn}}
{{eqn | l = \mathbf a^2
| r = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
| c =
}}
{{eqn | r =... | The [[Definition:Quaternion Group|Quaternion Group]] can be [[Definition:Generator of Group|generated]] by the [[Definition:Square Matrix|matrices]]:
:$\mathbf a = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}
\qquad
\mathbf b = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$
where $i$ is the [[Definition:Imaginary Uni... | Note that:
:$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
is the [[Definition:Identity Element|identity]] for [[Definition:Matrix Product (Conventional)|(conventional) matrix multiplication]] of [[Definition:Order of Square Matrix|order $2$]].
We have:
{{begin-eqn}}
{{eqn | l = \mathbf a^2
| r = \... | Generator for Quaternion Group | https://proofwiki.org/wiki/Generator_for_Quaternion_Group | https://proofwiki.org/wiki/Generator_for_Quaternion_Group | [
"Quaternion Group",
"Examples of Generators of Groups"
] | [
"Definition:Dicyclic Group/Quaternion Group",
"Definition:Generator of Group",
"Definition:Matrix/Square Matrix",
"Definition:Complex Number/Imaginary Unit"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Matrix Product (Conventional)",
"Definition:Matrix/Square Matrix/Order",
"Quaternion Group/Group Presentation",
"Category:Quaternion Group",
"Category:Examples of Generators of Groups"
] |
proofwiki-15230 | Group of Order 15 is Cyclic Group | Let $G$ be a group whose order is $15$.
Then $G$ is cyclic. | We have that $15 = 3 \times 5$.
Thus:
:$15$ is square-free
:$5 \equiv 2 \pmod 3$
:$3 \equiv 3 \pmod 5$
The conditions are fulfilled for Condition for Nu Function to be 1.
Thus $\map \nu {15} = 1$ and so all groups of order $15$ are cyclic.
{{Qed}} | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $15$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | We have that $15 = 3 \times 5$.
Thus:
:$15$ is [[Definition:Square-Free|square-free]]
:$5 \equiv 2 \pmod 3$
:$3 \equiv 3 \pmod 5$
The conditions are fulfilled for [[Condition for Nu Function to be 1]].
Thus $\map \nu {15} = 1$ and so all [[Definition:Group|groups]] of [[Definition:Order of Group|order]] $15$ are [[D... | Group of Order 15 is Cyclic Group/Proof 1 | https://proofwiki.org/wiki/Group_of_Order_15_is_Cyclic_Group | https://proofwiki.org/wiki/Group_of_Order_15_is_Cyclic_Group/Proof_1 | [
"Groups of Order 15",
"Finite Cyclic Groups",
"Group of Order 15 is Cyclic Group",
"Groups of Order p q"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Definition:Square-Free",
"Condition for Nu Function to be 1",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] |
proofwiki-15231 | Group of Order 15 is Cyclic Group | Let $G$ be a group whose order is $15$.
Then $G$ is cyclic. | From Number of Sylow p-Subgroups in Group of Order 15:
:the number of Sylow $3$-subgroups is in the set $\set {1, 4, 7, \ldots}$
:the number of Sylow $5$-subgroups is in the set $\set {1, 6, 11, \ldots}$.
From the Fifth Sylow Theorem
:the number of Sylow $3$-subgroups is a divisor of $15$
:the number of Sylow $5$-subgr... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $15$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | From [[Number of Sylow p-Subgroups in Group of Order 15]]:
:the number of [[Definition:Sylow p-Subgroup|Sylow $3$-subgroups]] is in the [[Definition:Set|set]] $\set {1, 4, 7, \ldots}$
:the number of [[Definition:Sylow p-Subgroup|Sylow $5$-subgroups]] is in the [[Definition:Set|set]] $\set {1, 6, 11, \ldots}$.
From the... | Group of Order 15 is Cyclic Group/Proof 2 | https://proofwiki.org/wiki/Group_of_Order_15_is_Cyclic_Group | https://proofwiki.org/wiki/Group_of_Order_15_is_Cyclic_Group/Proof_2 | [
"Groups of Order 15",
"Finite Cyclic Groups",
"Group of Order 15 is Cyclic Group",
"Groups of Order p q"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Number of Sylow p-Subgroups in Group of Order 15",
"Definition:Sylow p-Subgroup",
"Definition:Set",
"Definition:Sylow p-Subgroup",
"Definition:Set",
"Fifth Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Divisor (Algebra)/Integer",
"Definition:Sylow p-Subgroup",
"Definition:Divisor (Al... |
proofwiki-15232 | Group of Order 15 is Cyclic Group | Let $G$ be a group whose order is $15$.
Then $G$ is cyclic. | {{AimForCont}} $G$ is non-abelian.
Let $n_3$ denote the number of elements of $G$ of order $3$.
From Number of Elements of Order p in Group of Order pq is Multiple of q, $n_3$ is a multiple of $5$.
From Number of Order p Elements in Group with m Order p Subgroups, $n_3$ is a multiple of $2$.
Therefore $n_3$ is a multip... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $15$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | {{AimForCont}} $G$ is non-[[Definition:Abelian Group|abelian]].
Let $n_3$ denote the number of [[Definition:Element|elements]] of $G$ of [[Definition:Order of Group Element|order]] $3$.
From [[Number of Elements of Order p in Group of Order pq is Multiple of q]], $n_3$ is a [[Definition:Integer Multiple|multiple]] o... | Group of Order 15 is Cyclic Group/Proof 3 | https://proofwiki.org/wiki/Group_of_Order_15_is_Cyclic_Group | https://proofwiki.org/wiki/Group_of_Order_15_is_Cyclic_Group/Proof_3 | [
"Groups of Order 15",
"Finite Cyclic Groups",
"Group of Order 15 is Cyclic Group",
"Groups of Order p q"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Definition:Abelian Group",
"Definition:Element",
"Definition:Order of Group Element",
"Number of Elements of Order p in Group of Order pq is Multiple of q",
"Definition:Integral Multiple/Real Numbers",
"Number of Order p Elements in Group with m Order p Subgroups",
"Definition:Integral Multiple/Real Nu... |
proofwiki-15233 | Number of Abelian Groups | Let $n \in \Z_{\ge 1}$ be a (strictly) positive integer.
Let:
:$n = \ds \prod_{i \mathop = 1}^s p_i^{m_i}$
where the $p_i$ are distinct primes.
Let $\map {\nu_a} n$ denote the number of abelian groups of order $n$.
Then:
:$\map {\nu_a} n = \ds \prod_{i \mathop = 1}^s \map {\nu_a} {p_i^{m_i} }$
where:
:$\map {\nu_a} {p_... | {{ProofWanted|long and heavy proof which needs plenty work}} | Let $n \in \Z_{\ge 1}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let:
:$n = \ds \prod_{i \mathop = 1}^s p_i^{m_i}$
where the $p_i$ are [[Definition:Distinct Elements|distinct]] [[Definition:Prime Number|primes]].
Let $\map {\nu_a} n$ denote the number of [[Definition:Abelian Group|abe... | {{ProofWanted|long and heavy proof which needs plenty work}} | Number of Abelian Groups | https://proofwiki.org/wiki/Number_of_Abelian_Groups | https://proofwiki.org/wiki/Number_of_Abelian_Groups | [
"Abelian Groups"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Distinct/Plural",
"Definition:Prime Number",
"Definition:Abelian Group",
"Definition:Order of Structure",
"Definition:Integer Partition"
] | [] |
proofwiki-15234 | Order of Quotient Group | Let $G$ be a finite group.
Let $N$ be a normal subgroup of $G$.
Let $G / N$ be the quotient group of $G$ by $N$.
Then:
:$\dfrac {\order G} {\order N} = \order {G / N}$
where $\order G$ denotes the order of $G$. | From Lagrange's Theorem:
:$\dfrac {\order G} {\order N} = \index G N$
where $\index G N$ is the index of $N$ in $G$.
By definition of index:
:$\index G N = \order {G / N}$
{{qed}} | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $G / N$ be the [[Definition:Quotient Group|quotient group]] of $G$ by $N$.
Then:
:$\dfrac {\order G} {\order N} = \order {G / N}$
where $\order G$ denotes the [[Definition:Order of Group|or... | From [[Lagrange's Theorem (Group Theory)|Lagrange's Theorem]]:
:$\dfrac {\order G} {\order N} = \index G N$
where $\index G N$ is the [[Definition:Index of Subgroup|index]] of $N$ in $G$.
By definition of [[Definition:Index of Subgroup|index]]:
:$\index G N = \order {G / N}$
{{qed}} | Order of Quotient Group | https://proofwiki.org/wiki/Order_of_Quotient_Group | https://proofwiki.org/wiki/Order_of_Quotient_Group | [
"Quotient Groups",
"Order of Groups"
] | [
"Definition:Finite Group",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Order of Structure"
] | [
"Lagrange's Theorem (Group Theory)",
"Definition:Index of Subgroup",
"Definition:Index of Subgroup"
] |
proofwiki-15235 | Non-Abelian Simple Finite Groups are Infinitely Many | There exist infinitely many types of group which are non-abelian and finite. | We have that Alternating Group is Simple except on 4 Letters.
So for all $n \in \N$ such that $n \ne 4$, the alternating group $A_n$ is a simple group.
We also have that $A_n$ is non-abelian for all $n > 3$.
Hence the result.
{{qed}} | There exist [[Definition:Infinite Set|infinitely many]] [[Definition:Group Type|types of group]] which are non-[[Definition:Abelian Group|abelian]] and [[Definition:Finite Group|finite]]. | We have that [[Alternating Group is Simple except on 4 Letters]].
So for all $n \in \N$ such that $n \ne 4$, the [[Definition:Alternating Group|alternating group $A_n$]] is a [[Definition:Simple Group|simple group]].
We also have that $A_n$ is non-[[Definition:Abelian Group|abelian]] for all $n > 3$.
Hence the resul... | Non-Abelian Simple Finite Groups are Infinitely Many | https://proofwiki.org/wiki/Non-Abelian_Simple_Finite_Groups_are_Infinitely_Many | https://proofwiki.org/wiki/Non-Abelian_Simple_Finite_Groups_are_Infinitely_Many | [
"Simple Groups"
] | [
"Definition:Infinite Set",
"Definition:Group Type",
"Definition:Abelian Group",
"Definition:Finite Group"
] | [
"Alternating Group is Simple except on 4 Letters",
"Definition:Alternating Group",
"Definition:Simple Group",
"Definition:Abelian Group"
] |
proofwiki-15236 | Centralizer of Self-Inverse Element of Non-Abelian Finite Simple Group is not That Group | Let $G$ be a non-abelian finite simple group.
Let $t \in G$ be a self-inverse element of $G$.
Then:
:$\map {C_G} t \ne G$
where $\map {C_G} t$ denotes the centralizer of $t$ in $G$. | Let $G$ be a non-abelian finite simple group.
Let $t \in G$ which is not the identity.
By definition of a simple group and Center of Group is Normal Subgroup:
:either $\map Z G = G$ or $\map Z G$ is the trivial group.
By definition of an abelian group:
:$\map Z G = G$ {{iff}} $G$ is abelian
Hence we must have $\map Z G... | Let $G$ be a non-[[Definition:Abelian Group|abelian]] [[Definition:Finite Group|finite]] [[Definition:Simple Group|simple group]].
Let $t \in G$ be a [[Definition:Self-Inverse Element|self-inverse element]] of $G$.
Then:
:$\map {C_G} t \ne G$
where $\map {C_G} t$ denotes the [[Definition:Centralizer|centralizer]] of... | Let $G$ be a non-[[Definition:Abelian Group|abelian]] [[Definition:Finite Group|finite]] [[Definition:Simple Group|simple group]].
Let $t \in G$ which is not the [[Definition:Identity Element|identity]].
By definition of a [[Definition:Simple Group|simple group]] and [[Center of Group is Normal Subgroup]]:
:either $... | Centralizer of Self-Inverse Element of Non-Abelian Finite Simple Group is not That Group | https://proofwiki.org/wiki/Centralizer_of_Self-Inverse_Element_of_Non-Abelian_Finite_Simple_Group_is_not_That_Group | https://proofwiki.org/wiki/Centralizer_of_Self-Inverse_Element_of_Non-Abelian_Finite_Simple_Group_is_not_That_Group | [
"Centralizers",
"Simple Groups",
"Self-Inverse Elements",
"Finite Groups"
] | [
"Definition:Abelian Group",
"Definition:Finite Group",
"Definition:Simple Group",
"Definition:Self-Inverse Element",
"Definition:Centralizer"
] | [
"Definition:Abelian Group",
"Definition:Finite Group",
"Definition:Simple Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Simple Group",
"Center of Group is Normal Subgroup",
"Definition:Trivial Group",
"Definition:Abelian Group/Definition 2",
"Definition:Trivial Gro... |
proofwiki-15237 | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 1 | :$\norm {\, \cdot \,}_1$ is well-defined.
That is,
:$(1): \quad \ds \forall \eqclass {x_n}{}: \lim_{n \mathop \to \infty} \norm{x_n}$ exists
:$(2): \quad \ds \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \CC \,\big / \NN: \eqclass {x_n}{} = \eqclass {y_n}{} \implies \lim_{n \mathop \to \infty} \norm{x_n} = \lim_{n \m... | By Norm Sequence of Cauchy Sequence has Limit then:
:for each $\eqclass {x_n}{}$ the $\ds \lim_{n \mathop \to \infty} \norm{x_n}$ exists.
Suppose $\eqclass {x_n}{} = \eqclass {y_n}{}$.
By Left Cosets are Equal iff Difference in Subgroup then:
:$\sequence {x_n} - \sequence {y_n} = \sequence {x_n - y_n} \in \NN$
By Equi... | :$\norm {\, \cdot \,}_1$ is [[Definition:Well-Defined Mapping|well-defined]].
That is,
:$(1): \quad \ds \forall \eqclass {x_n}{}: \lim_{n \mathop \to \infty} \norm{x_n}$ exists
:$(2): \quad \ds \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \CC \,\big / \NN: \eqclass {x_n}{} = \eqclass {y_n}{} \implies \lim_{n \matho... | By [[Norm Sequence of Cauchy Sequence has Limit]] then:
:for each $\eqclass {x_n}{}$ the $\ds \lim_{n \mathop \to \infty} \norm{x_n}$ exists.
Suppose $\eqclass {x_n}{} = \eqclass {y_n}{}$.
By [[Left Cosets are Equal iff Product with Inverse in Subgroup|Left Cosets are Equal iff Difference in Subgroup]] then:
:$\seque... | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 1 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_1 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_1 | [
"Completion of Normed Division Ring"
] | [
"Definition:Well-Defined/Mapping"
] | [
"Norm Sequence of Cauchy Sequence has Limit",
"Left Cosets are Equal iff Product with Inverse in Subgroup",
"Equivalent Cauchy Sequences have Equal Limits of Norm Sequences"
] |
proofwiki-15238 | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 2 | :$\norm {\, \cdot \,}_1$ satisfies {{Norm-axiom-mult|1}}
That is:
:$\forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = 0 \iff \eqclass {x_n} {} = \eqclass {0_R} {} $ | By Quotient Ring of Cauchy Sequences is Division Ring the zero of $\CC \,\big / \NN$ is $\eqclass {0_R} {}$.
{{begin-eqn}}
{{eqn | l = \norm {\eqclass {0_R} {} }_1 = 0
| o = \leadstoandfrom
| r = \lim_{n \mathop \to \infty} \norm {x_n} = 0
| c = Definition of $\norm {\,\cdot\,}_1$
}}
{{eqn | o = \lead... | :$\norm {\, \cdot \,}_1$ satisfies {{Norm-axiom-mult|1}}
That is:
:$\forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = 0 \iff \eqclass {x_n} {} = \eqclass {0_R} {} $ | By [[Quotient Ring of Cauchy Sequences is Division Ring]] the [[Definition:Ring Zero|zero]] of $\CC \,\big / \NN$ is $\eqclass {0_R} {}$.
{{begin-eqn}}
{{eqn | l = \norm {\eqclass {0_R} {} }_1 = 0
| o = \leadstoandfrom
| r = \lim_{n \mathop \to \infty} \norm {x_n} = 0
| c = Definition of $\norm {\,\c... | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 2 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_2 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_2 | [
"Completion of Normed Division Ring"
] | [] | [
"Quotient Ring of Cauchy Sequences is Division Ring",
"Definition:Ring Zero",
"Left Cosets are Equal iff Product with Inverse in Subgroup"
] |
proofwiki-15239 | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 3 | :$\norm {\, \cdot \,}_1$ satisfies the {{Norm-axiom-mult|2}}.
That is:
:$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} \eqclass {y_n} {} }_1 = \norm {\eqclass {x_n} {} }_1 \times \norm {\eqclass {y_n} {} }_1$ | Let $\eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN$
{{begin-eqn}}
{{eqn | l = \norm {\eqclass {x_n} {} \eqclass {y_n} {} }_1
| r = \norm {\eqclass {x_n y_n} {} }_1
| c = Multiplication on quotient ring
}}
{{eqn | r = \lim_{n \mathop \to \infty} \norm {x_n y_n}
| c = Definition of $\norm {\... | :$\norm {\, \cdot \,}_1$ satisfies the {{Norm-axiom-mult|2}}.
That is:
:$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} \eqclass {y_n} {} }_1 = \norm {\eqclass {x_n} {} }_1 \times \norm {\eqclass {y_n} {} }_1$ | Let $\eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN$
{{begin-eqn}}
{{eqn | l = \norm {\eqclass {x_n} {} \eqclass {y_n} {} }_1
| r = \norm {\eqclass {x_n y_n} {} }_1
| c = Multiplication on quotient ring
}}
{{eqn | r = \lim_{n \mathop \to \infty} \norm {x_n y_n}
| c = Definition of $\norm {... | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 3 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_3 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_3 | [
"Completion of Normed Division Ring"
] | [] | [
"Combination Theorem for Sequences/Real/Product Rule"
] |
proofwiki-15240 | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 4 | :$\norm {\, \cdot \,}_1$ satisfies the {{Norm-axiom-mult|3}}.
That is:
:$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} + \eqclass {y_n} {} }_1 \le \norm {\eqclass {x_n} {} }_1 + \norm {\eqclass {y_n} {} }_1$ | Let $\eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN$
{{begin-eqn}}
{{eqn | l = \norm {\eqclass {x_n} {} + \eqclass {y_n} {} } _1
| r = \norm {\eqclass {x_n + y_n} {} }_1
| c = Addition on quotient ring
}}
{{eqn | r = \lim_{n \mathop \to \infty} \norm {x_n + y_n}
| c = Definition of $\norm {... | :$\norm {\, \cdot \,}_1$ satisfies the {{Norm-axiom-mult|3}}.
That is:
:$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} + \eqclass {y_n} {} }_1 \le \norm {\eqclass {x_n} {} }_1 + \norm {\eqclass {y_n} {} }_1$ | Let $\eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN$
{{begin-eqn}}
{{eqn | l = \norm {\eqclass {x_n} {} + \eqclass {y_n} {} } _1
| r = \norm {\eqclass {x_n + y_n} {} }_1
| c = Addition on quotient ring
}}
{{eqn | r = \lim_{n \mathop \to \infty} \norm {x_n + y_n}
| c = Definition of $\norm ... | Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 4 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_4 | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring/Lemma_4 | [
"Completion of Normed Division Ring"
] | [] | [
"Inequality Rule for Real Sequences",
"Combination Theorem for Sequences/Real/Sum Rule"
] |
proofwiki-15241 | Invertible Elements of Monoid form Subgroup | Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.
Let $U \subseteq S$ be the subset of $S$ consisting of the invertible elements of $S$.
Then $\struct {U, \circ}$ forms a subgroup of $S$. | We have from Inverse of Identity Element is Itself that $e$ is invertible.
Hence $e \in U$ and so $U \ne \O$.
Let $x, y \in U$.
As $x$ and $y$ are invertible, it follows that $x^{-1}$ and $y^{-1}$ both exist in $S$.
Both $x^{-1}$ and $y^{-1}$ also have inverses $x$ and $y$ respectively, and so themselves are invertible... | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $U \subseteq S$ be the [[Definition:Subset|subset]] of $S$ consisting of the [[Definition:Invertible Element|invertible elements]] of $S$.
Then $\struct {U, \circ}$ forms a [[Definition:Subgr... | We have from [[Inverse of Identity Element is Itself]] that $e$ is [[Definition:Invertible Element|invertible]].
Hence $e \in U$ and so $U \ne \O$.
Let $x, y \in U$.
As $x$ and $y$ are [[Definition:Invertible Element|invertible]], it follows that $x^{-1}$ and $y^{-1}$ both exist in $S$.
Both $x^{-1}$ and $y^{-1}$ ... | Invertible Elements of Monoid form Subgroup | https://proofwiki.org/wiki/Invertible_Elements_of_Monoid_form_Subgroup | https://proofwiki.org/wiki/Invertible_Elements_of_Monoid_form_Subgroup | [
"Monoids",
"Subgroups",
"Inverse Elements"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subset",
"Definition:Invertible Element",
"Definition:Subgroup"
] | [
"Inverse of Identity Element is Itself",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Invertible Element",
"Definition:Associative Operation",
"Definition:Invertible Element",
"Two-Step Subgroup Test"
] |
proofwiki-15242 | Combination Theorem for Sequences/Normed Division Ring/Inverse Rule/Lemma | :$\ds \lim_{n \mathop \to \infty} {y_n}^{-1} = l^{-1}$ | By Limit of Subsequence equals Limit of Sequence then $\sequence {y_n}$ is convergent with:
:$\ds \lim_{n \mathop \to \infty} y_n = l$
Let $\epsilon > 0$ be given.
Let $\epsilon' = \dfrac {\epsilon {\norm l}^2 } {2}$.
Then:
:$ \epsilon' > 0$
As $\sequence {y_n} \to l$, as $n \to \infty$, we can find $N_1$ such that:
:... | :$\ds \lim_{n \mathop \to \infty} {y_n}^{-1} = l^{-1}$ | By [[Limit of Subsequence equals Limit of Sequence/Normed Division Ring|Limit of Subsequence equals Limit of Sequence]] then $\sequence {y_n}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] with:
:$\ds \lim_{n \mathop \to \infty} y_n = l$
Let $\epsilon > 0$ be given.
Let $\epsilon' = \dfrac... | Combination Theorem for Sequences/Normed Division Ring/Inverse Rule/Lemma | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Inverse_Rule/Lemma | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Normed_Division_Ring/Inverse_Rule/Lemma | [
"Combination Theorem for Sequences in Normed Division Rings"
] | [] | [
"Limit of Subsequence equals Limit of Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Sequence Converges to Within Half Limit/Normed Division Ring",
"Definition:Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Category:Combination Theorem for ... |
proofwiki-15243 | Division Ring Norm is Continuous on Induced Metric Space | Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
The mapping $\norm {\,\cdot\,} : \struct {R, d} \to \R$ is continuous. | Let $x_0 \in R$.
Let $\epsilon \in \R_{>0}$.
Let $x \in R: \norm {x - x_0} < \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \size {\norm x - \norm {x_0} }
| o = \le
| r = \norm {x - x_0}
| c = Reverse Triangle Inequality on Normed Division Ring
}}
{{eqn | o = <
| r = \epsilon
}}
{{end-eqn}}
By the ... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $d$ be the [[Definition:Metric Induced by Norm|metric induced]] by the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
The [[Definition:Mapping|mapping]] $\norm {\,\cdot\,} : \struct {R, d} \to \R$ i... | Let $x_0 \in R$.
Let $\epsilon \in \R_{>0}$.
Let $x \in R: \norm {x - x_0} < \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \size {\norm x - \norm {x_0} }
| o = \le
| r = \norm {x - x_0}
| c = [[Reverse Triangle Inequality on Normed Division Ring]]
}}
{{eqn | o = <
| r = \epsilon
}}
{{end-eqn}}
... | Division Ring Norm is Continuous on Induced Metric Space | https://proofwiki.org/wiki/Division_Ring_Norm_is_Continuous_on_Induced_Metric_Space | https://proofwiki.org/wiki/Division_Ring_Norm_is_Continuous_on_Induced_Metric_Space | [
"Normed Division Rings",
"Norm Theory"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm",
"Definition:Norm/Division Ring",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Reverse Triangle Inequality/Normed Division Ring",
"Definition:Metric Induced by Norm",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Category:Normed Division Rings",
"Category:Norm Theory"
] |
proofwiki-15244 | Sum of Sequence of Squares of Fibonacci Numbers | :$\forall n \ge 1: \ds \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$
That is:
:${F_1}^2 + {F_2}^2 + {F_3}^2 + \cdots + {F_n}^2 = F_n F_{n + 1}$ | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$ | :$\forall n \ge 1: \ds \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$
That is:
:${F_1}^2 + {F_2}^2 + {F_3}^2 + \cdots + {F_n}^2 = F_n F_{n + 1}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$ | Sum of Sequence of Squares of Fibonacci Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Squares_of_Fibonacci_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Squares_of_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Sums of Sequences",
"Proofs by Induction"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-15245 | Properties of Norm on Division Ring/Norm of Negative | :$\norm {-x} = \norm x$ | By Norm of Negative of Unity:
:$\norm {-1_R} = 1$
Then:
{{begin-eqn}}
{{eqn | l = \norm {-x}
| r = \norm {-1_R \circ x}
| c = Product with Ring Negative
}}
{{eqn | r = \norm {-1_R} \norm x
| c = {{Norm-axiom-mult|2}}
}}
{{eqn | r = \norm x
| c = Norm of Negative of Unity
}}
{{end-eqn}}
as desire... | :$\norm {-x} = \norm x$ | By [[Properties of Norm on Division Ring/Norm of Negative of Unity|Norm of Negative of Unity]]:
:$\norm {-1_R} = 1$
Then:
{{begin-eqn}}
{{eqn | l = \norm {-x}
| r = \norm {-1_R \circ x}
| c = [[Product with Ring Negative]]
}}
{{eqn | r = \norm {-1_R} \norm x
| c = {{Norm-axiom-mult|2}}
}}
{{eqn | r... | Properties of Norm on Division Ring/Norm of Negative | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Negative | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Negative | [
"Properties of Norm on Division Ring"
] | [] | [
"Properties of Norm on Division Ring/Norm of Negative of Unity",
"Product with Ring Negative",
"Properties of Norm on Division Ring/Norm of Negative of Unity"
] |
proofwiki-15246 | Properties of Norm on Division Ring/Norm of Unity | :$\norm {1_R} = 1$. | By {{Norm-axiom-mult|2}}:
:$\forall x, y \in R: \norm {x \circ y} = \norm x \norm y$
In particular:
:$\norm {1_R} = \norm {1_R \circ 1_R} = \norm {1_R} \norm {1_R}$
By {{Norm-axiom-mult|1}}:
:$\norm {1_R} \ne 0$
So $\norm {1_R}$ has an inverse in $R$.
Multiplying by this inverse:
:$\norm {1_R} \norm {1_R} = \norm {1_R}... | :$\norm {1_R} = 1$. | By {{Norm-axiom-mult|2}}:
:$\forall x, y \in R: \norm {x \circ y} = \norm x \norm y$
In particular:
:$\norm {1_R} = \norm {1_R \circ 1_R} = \norm {1_R} \norm {1_R}$
By {{Norm-axiom-mult|1}}:
:$\norm {1_R} \ne 0$
So $\norm {1_R}$ has an [[Definition:Multiplicative Inverse|inverse]] in $R$.
Multiplying by this [[D... | Properties of Norm on Division Ring/Norm of Unity | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Unity | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Unity | [
"Properties of Norm on Division Ring"
] | [] | [
"Definition:Multiplicative Inverse",
"Definition:Multiplicative Inverse"
] |
proofwiki-15247 | Properties of Norm on Division Ring/Norm of Negative of Unity | :$\norm {-1_R} = 1$ | By Product of Ring Negatives:
:$-1_R \circ -1_R = 1_R \circ 1_R = 1_R$
So:
{{begin-eqn}}
{{eqn | l = \norm {-1_R}^2
| r = \norm {-1_R} \norm {-1_R}
}}
{{eqn | r = \norm {-1_R \circ -1_R}
| c = {{Norm-axiom-mult|2}}
}}
{{eqn | r = \norm {1_R}
| c = Product of Ring Negatives
}}
{{eqn | r = 1
| c =... | :$\norm {-1_R} = 1$ | By [[Product of Ring Negatives]]:
:$-1_R \circ -1_R = 1_R \circ 1_R = 1_R$
So:
{{begin-eqn}}
{{eqn | l = \norm {-1_R}^2
| r = \norm {-1_R} \norm {-1_R}
}}
{{eqn | r = \norm {-1_R \circ -1_R}
| c = {{Norm-axiom-mult|2}}
}}
{{eqn | r = \norm {1_R}
| c = [[Product of Ring Negatives]]
}}
{{eqn | r = 1
... | Properties of Norm on Division Ring/Norm of Negative of Unity | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Negative_of_Unity | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Negative_of_Unity | [
"Properties of Norm on Division Ring"
] | [] | [
"Product of Ring Negatives",
"Product of Ring Negatives",
"Properties of Norm on Division Ring/Norm of Unity"
] |
proofwiki-15248 | Repunit is Zuckerman Number | Let $n$ be a repunit.
Then $n$ is also a Zuckerman number. | The digits of a repunit are by definition all $1$.
Thus the product of the digits of a repunit is $1$.
By One Divides all Integers, $1$ is a divisor of $n$.
Hence the result, by definition of Zuckerman number. | Let $n$ be a [[Definition:Repunit|repunit]].
Then $n$ is also a [[Definition:Zuckerman Number|Zuckerman number]]. | The [[Definition:Digit|digits]] of a [[Definition:Repunit|repunit]] are by definition all $1$.
Thus the [[Definition:Integer Multiplication|product]] of the [[Definition:Digit|digits]] of a [[Definition:Repunit|repunit]] is $1$.
By [[One Divides all Integers]], $1$ is a [[Definition:Divisor|divisor]] of $n$.
Hence t... | Repunit is Zuckerman Number | https://proofwiki.org/wiki/Repunit_is_Zuckerman_Number | https://proofwiki.org/wiki/Repunit_is_Zuckerman_Number | [
"Repunits",
"Zuckerman Numbers"
] | [
"Definition:Repunit",
"Definition:Zuckerman Number"
] | [
"Definition:Digit",
"Definition:Repunit",
"Definition:Multiplication/Integers",
"Definition:Digit",
"Definition:Repunit",
"Integer Divisor Results/One Divides all Integers",
"Definition:Divisor",
"Definition:Zuckerman Number"
] |
proofwiki-15249 | Properties of Norm on Division Ring/Norm of Difference | :$\norm {x - y} \le \norm x + \norm y$ | Then:
{{begin-eqn}}
{{eqn | l = \norm {x - y}
| r = \norm {x + \paren {-y} }
}}
{{eqn | o = \le
| r = \norm x + \norm {-y}
| c = {{Norm-axiom-mult|3}}
}}
{{eqn | r = \norm x + \norm y
| c = Norm of Ring Negative
}}
{{end-eqn}}
as desired.
{{qed}} | :$\norm {x - y} \le \norm x + \norm y$ | Then:
{{begin-eqn}}
{{eqn | l = \norm {x - y}
| r = \norm {x + \paren {-y} }
}}
{{eqn | o = \le
| r = \norm x + \norm {-y}
| c = {{Norm-axiom-mult|3}}
}}
{{eqn | r = \norm x + \norm y
| c = [[Norm of Ring Negative]]
}}
{{end-eqn}}
as desired.
{{qed}} | Properties of Norm on Division Ring/Norm of Difference | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Difference | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Difference | [
"Properties of Norm on Division Ring"
] | [] | [
"Properties of Norm on Division Ring/Norm of Negative"
] |
proofwiki-15250 | Properties of Norm on Division Ring/Norm of Inverse | :$x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$ | Let $x \ne 0_R$.
By {{Norm-axiom-mult|1}}:
:$\norm x \ne 0$
So:
{{begin-eqn}}
{{eqn| l = \norm x \norm {x^{-1} }
| r = \norm {x \circ x^{-1} }
| c = {{Norm-axiom-mult|2}}
}}
{{eqn| r = \norm {1_R}
| c = {{Defof|Product Inverse}}
}}
{{eqn| r = 1
| c = Norm of Unity of Division Ring
}}
{{eqn| ll= \lea... | :$x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$ | Let $x \ne 0_R$.
By {{Norm-axiom-mult|1}}:
:$\norm x \ne 0$
So:
{{begin-eqn}}
{{eqn| l = \norm x \norm {x^{-1} }
| r = \norm {x \circ x^{-1} }
| c = {{Norm-axiom-mult|2}}
}}
{{eqn| r = \norm {1_R}
| c = {{Defof|Product Inverse}}
}}
{{eqn| r = 1
| c = [[Norm of Unity of Division Ring]]
}}
{{eqn| ll... | Properties of Norm on Division Ring/Norm of Inverse | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Inverse | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Inverse | [
"Properties of Norm on Division Ring"
] | [] | [
"Properties of Norm on Division Ring/Norm of Unity"
] |
proofwiki-15251 | Properties of Norm on Division Ring/Norm of Quotient | :$y \ne 0_R \implies \norm {x y^{-1} } = \norm {y^{-1} x} = \dfrac {\norm x} {\norm y}$ | Let $y \ne 0_R$.
By {{Norm-axiom-mult|1}} then:
:$\norm y \ne 0$
So:
{{begin-eqn}}
{{eqn| l = \norm {x \circ y^{-1} }
| r = \norm x \norm {y^{-1} }
| c = {{Norm-axiom-mult|2}}
}}
{{eqn| r = \dfrac {\norm x} {\norm y}
| c = Norm of Inverse
}}
{{end-eqn}}
Similarly:
:$\norm {y^{-1} x} = \dfrac {\norm x} {\... | :$y \ne 0_R \implies \norm {x y^{-1} } = \norm {y^{-1} x} = \dfrac {\norm x} {\norm y}$ | Let $y \ne 0_R$.
By {{Norm-axiom-mult|1}} then:
:$\norm y \ne 0$
So:
{{begin-eqn}}
{{eqn| l = \norm {x \circ y^{-1} }
| r = \norm x \norm {y^{-1} }
| c = {{Norm-axiom-mult|2}}
}}
{{eqn| r = \dfrac {\norm x} {\norm y}
| c = [[Properties of Norm on Division Ring/Norm of Inverse|Norm of Inverse]]
}}
{{end... | Properties of Norm on Division Ring/Norm of Quotient | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Quotient | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Quotient | [
"Properties of Norm on Division Ring"
] | [] | [
"Properties of Norm on Division Ring/Norm of Inverse"
] |
proofwiki-15252 | Properties of Norm on Division Ring/Norm of Power Equals Unity | :$\forall n \in \N_{>0}: \norm {x^n} = 1 \implies \norm x = 1$ | Let $n \in \N_{>0}$.
Let $\norm {x^n} = 1$.
By {{Norm-axiom-mult|2}}:
:$\norm x^n = 1$
Since $\norm x \ge 0$, by Positive Real Complex Root of Unity:
:$\norm x = 1$
as desired.
{{qed}} | :$\forall n \in \N_{>0}: \norm {x^n} = 1 \implies \norm x = 1$ | Let $n \in \N_{>0}$.
Let $\norm {x^n} = 1$.
By {{Norm-axiom-mult|2}}:
:$\norm x^n = 1$
Since $\norm x \ge 0$, by [[Positive Real Complex Root of Unity]]:
:$\norm x = 1$
as desired.
{{qed}} | Properties of Norm on Division Ring/Norm of Power Equals Unity | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Power_Equals_Unity | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Power_Equals_Unity | [
"Properties of Norm on Division Ring"
] | [] | [
"Positive Real Complex Root of Unity"
] |
proofwiki-15253 | Properties of Norm on Division Ring/Norm of Integer | For all $n \in \N_{>0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
Then:
:$\norm {n \cdot 1_R} \le n$. | Let $n \in \N_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {n \cdot 1_R}
| r = \norm {1_R + 1_R + \dots + 1_R}
}}
{{eqn | o = \le
| r = \underbrace {\norm {1_R} + \norm {1_R} + \dots + \norm {1_R} }_{\text {$n$ times} }
| c = {{Norm-axiom-mult|3}}
}}
{{eqn | r = \underbrace {1 + 1 + \dots + 1 }_{\text... | For all $n \in \N_{>0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
Then:
:$\norm {n \cdot 1_R} \le n$. | Let $n \in \N_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {n \cdot 1_R}
| r = \norm {1_R + 1_R + \dots + 1_R}
}}
{{eqn | o = \le
| r = \underbrace {\norm {1_R} + \norm {1_R} + \dots + \norm {1_R} }_{\text {$n$ times} }
| c = {{Norm-axiom-mult|3}}
}}
{{eqn | r = \underbrace {1 + 1 + \dots + 1 }_{\tex... | Properties of Norm on Division Ring/Norm of Integer | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Integer | https://proofwiki.org/wiki/Properties_of_Norm_on_Division_Ring/Norm_of_Integer | [
"Properties of Norm on Division Ring"
] | [] | [
"Properties of Norm on Division Ring/Norm of Unity"
] |
proofwiki-15254 | Conversion from Hexadecimal to Binary | Let $n$ be a (positive) integer expressed in hexadecimal notation as:
:$n = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0}_H$
Then $n$ can be expressed in binary notation as:
:$n = \sqbrk {b_{r 3} b_{r 2} b_{r 1} b_{r 0} b_{\paren {r - 1} 3} b_{\paren {r - 1} 2} b_{\paren {r - 1} 1} b_{\paren {r - 1} 0} \dotso b_{1 3} b_{1 2} b... | We have:
{{begin-eqn}}
{{eqn | l = n
| r = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0}_H
| c =
}}
{{eqn | r = \sum_{j \mathop = 0}^r a_j 16^j
| c = {{Defof|Hexadecimal Notation}}
}}
{{end-eqn}}
We have that:
:$0 \le a_j < 16$
and so:
{{begin-eqn}}
{{eqn | l = a_j
| r = \sqbrk {b_{j 3} b_{j 2} b_{j 1}... | Let $n$ be a [[Definition:Positive Integer|(positive) integer]] expressed in [[Definition:Hexadecimal Notation|hexadecimal notation]] as:
:$n = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0}_H$
Then $n$ can be expressed in [[Definition:Binary Notation|binary notation]] as:
:$n = \sqbrk {b_{r 3} b_{r 2} b_{r 1} b_{r 0} b_{\p... | We have:
{{begin-eqn}}
{{eqn | l = n
| r = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0}_H
| c =
}}
{{eqn | r = \sum_{j \mathop = 0}^r a_j 16^j
| c = {{Defof|Hexadecimal Notation}}
}}
{{end-eqn}}
We have that:
:$0 \le a_j < 16$
and so:
{{begin-eqn}}
{{eqn | l = a_j
| r = \sqbrk {b_{j 3} b_{j 2} b_... | Conversion from Hexadecimal to Binary | https://proofwiki.org/wiki/Conversion_from_Hexadecimal_to_Binary | https://proofwiki.org/wiki/Conversion_from_Hexadecimal_to_Binary | [
"Hexadecimal Notation",
"Binary Notation",
"Conversion from Hexadecimal to Binary"
] | [
"Definition:Positive/Integer",
"Definition:Hexadecimal Notation",
"Definition:Binary Notation",
"Definition:Hexadecimal Notation",
"Definition:Binary Notation",
"Definition:Binary Notation",
"Definition:Hexadecimal Notation",
"Definition:Zero Digit",
"Definition:Bit",
"Definition:Concatenation (Fo... | [] |
proofwiki-15255 | Birthday Paradox/General/3 | Let $n$ be a set of people.
Let the probability that at least $3$ of them have the same birthday be greater than $50 \%$.
Then $n \ge 88$. | Let $\map F {r, n}$ be the number of ways to distribute $r$ objects into $n$ cells such that there are no more than $2$ objects in each cell.
Let there be $d$ cells which are each occupied by $2$ objects.
These can be chosen in $\dbinom n d$ ways.
There remain $s = r - 2 d$ objects which can then be distributed among $... | Let $n$ be a [[Definition:Set|set]] of people.
Let the [[Definition:Probability|probability]] that at least $3$ of them have the same birthday be greater than $50 \%$.
Then $n \ge 88$. | Let $\map F {r, n}$ be the number of ways to distribute $r$ objects into $n$ cells such that there are no more than $2$ objects in each cell.
Let there be $d$ cells which are each occupied by $2$ objects.
These can be chosen in $\dbinom n d$ ways.
There remain $s = r - 2 d$ objects which can then be distributed amon... | Birthday Paradox/General/3 | https://proofwiki.org/wiki/Birthday_Paradox/General/3 | https://proofwiki.org/wiki/Birthday_Paradox/General/3 | [
"Birthday Paradox"
] | [
"Definition:Set",
"Definition:Probability"
] | [
"Definition:Permutation",
"Definition:Probability",
"Definition:Probability"
] |
proofwiki-15256 | Convergent Sequence is Cauchy Sequence/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring.
Every convergent sequence in $R$ is a Cauchy sequence. | Let $\sequence {x_n}$ be a sequence in $R$ that converges to the limit $l \in R$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ converges to $l$, we have:
:$\exists N: \forall n > N: \norm {x_n - l} < \dfrac \epsilon 2$
So if $m > N$ and $n > N$, then:
{{begin-eqn}}
{{eqn | l = \norm... | Let $\struct {R, \norm {\,\cdot\,}} $ be a [[Definition:Normed Division Ring|normed division ring]].
Every [[Definition:Convergent Sequence in Normed Division Ring|convergent sequence]] in $R$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $R$ that [[Definition:Convergent Sequence in Normed Division Ring|converges]] to the [[Definition:Limit of Sequence (Normed Division Ring)|limit]] $l \in R$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ [[Definition:C... | Convergent Sequence is Cauchy Sequence/Normed Division Ring/Proof 1 | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Normed_Division_Ring | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Normed_Division_Ring/Proof_1 | [
"Convergent Sequence in Normed Division Ring is Cauchy Sequence",
"Convergent Sequence is Cauchy Sequence",
"Convergent Sequences in Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-15257 | Convergent Sequence is Cauchy Sequence/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring.
Every convergent sequence in $R$ is a Cauchy sequence. | Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a sequence in $R$ that converges to the limit $l$ in $\struct {R, \norm {\,\cdot\,}}$.
Thus, by definition, $\sequence {x_n} $ converges to the limit $l$ in $\struct {R, d}$.
By Convergent Sequence is Cauchy Sequence in metri... | Let $\struct {R, \norm {\,\cdot\,}} $ be a [[Definition:Normed Division Ring|normed division ring]].
Every [[Definition:Convergent Sequence in Normed Division Ring|convergent sequence]] in $R$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] on $R$ be the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $R$ that [[Definition:Convergent Sequence in Normed Division Ring|converges]] to the [[Definit... | Convergent Sequence is Cauchy Sequence/Normed Division Ring/Proof 2 | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Normed_Division_Ring | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Normed_Division_Ring/Proof_2 | [
"Convergent Sequence in Normed Division Ring is Cauchy Sequence",
"Convergent Sequence is Cauchy Sequence",
"Convergent Sequences in Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Limit of Sequence/N... |
proofwiki-15258 | Sum of 3 Squares in 2 Distinct Ways | $27$ is the smallest positive integer which can be expressed as the sum of $3$ square numbers in $2$ distinct ways:
{{begin-eqn}}
{{eqn | l = 27
| r = 3^2 + 3^2 + 3^2
}}
{{eqn | r = 5^2 + 1^2 + 1^2
}}
{{end-eqn}} | Can be performed by brute-force investigation. | $27$ is the smallest [[Definition:Positive Integer|positive integer]] which can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Square Number|square numbers]] in $2$ [[Definition:Distinct|distinct]] ways:
{{begin-eqn}}
{{eqn | l = 27
| r = 3^2 + 3^2 + 3^2
}}
{{eqn | r = 5^2 + 1^2 + 1^... | Can be performed by brute-force investigation. | Sum of 3 Squares in 2 Distinct Ways | https://proofwiki.org/wiki/Sum_of_3_Squares_in_2_Distinct_Ways | https://proofwiki.org/wiki/Sum_of_3_Squares_in_2_Distinct_Ways | [
"Square Numbers",
"27"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Distinct"
] | [] |
proofwiki-15259 | Triangular Numbers which are Sum of Two Cubes | The sequence of triangular numbers which are the sum of $2$ cubes begins:
:$28, 91, 351, 2926, 8001, 46971, 58653, 93528, 97461, \dots$
{{OEIS|A113958}} | Can be demonstrated by brute force.
For example:
{{begin-eqn}}
{{eqn | l = 28
| r = 1 + 27
| c =
}}
{{eqn | r = 1^3 + 3^3
| c =
}}
{{eqn | r = \dfrac {7 \paren {7 + 1} } 2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 91
| r = 27 + 64
| c =
}}
{{eqn | r = 3^3 + 4^3
| c =
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Triangular Number|triangular numbers]] which are the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Cube Number|cubes]] begins:
:$28, 91, 351, 2926, 8001, 46971, 58653, 93528, 97461, \dots$
{{OEIS|A113958}} | Can be demonstrated by brute force.
For example:
{{begin-eqn}}
{{eqn | l = 28
| r = 1 + 27
| c =
}}
{{eqn | r = 1^3 + 3^3
| c =
}}
{{eqn | r = \dfrac {7 \paren {7 + 1} } 2
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 91
| r = 27 + 64
| c =
}}
{{eqn | r = 3^3 + 4^3
| c... | Triangular Numbers which are Sum of Two Cubes | https://proofwiki.org/wiki/Triangular_Numbers_which_are_Sum_of_Two_Cubes | https://proofwiki.org/wiki/Triangular_Numbers_which_are_Sum_of_Two_Cubes | [
"Cube Numbers",
"Triangular Numbers",
"28"
] | [
"Definition:Integer Sequence",
"Definition:Triangular Number",
"Definition:Addition/Integers",
"Definition:Cube Number"
] | [] |
proofwiki-15260 | Product of Factors of Even Perfect Number | Let $P$ be the perfect number $2^{n - 1} \paren {2^n - 1}$.
Then:
:$\ds \prod_{d \mathop \divides P} d = P^n$ | The factors of $P$ are:
:$1, 2, 4, \dots, 2^{n - 1}, 2^n - 1, 2 \paren {2^n - 1}, \dots, 2^{n - 1} \paren {2^n - 1}$
Therefore their product is:
{{begin-eqn}}
{{eqn | l = \prod_{d \mathop \divides P} d
| r = \paren {\prod_{i \mathop = 0}^{n - 1} 2^i} \paren {\prod_{i \mathop = 0}^{n - 1} 2^i \paren {2^n - 1} }
}}... | Let $P$ be the [[Definition:Perfect Number|perfect number]] $2^{n - 1} \paren {2^n - 1}$.
Then:
:$\ds \prod_{d \mathop \divides P} d = P^n$ | The [[Definition:Divisor of Integer|factors]] of $P$ are:
:$1, 2, 4, \dots, 2^{n - 1}, 2^n - 1, 2 \paren {2^n - 1}, \dots, 2^{n - 1} \paren {2^n - 1}$
Therefore their product is:
{{begin-eqn}}
{{eqn | l = \prod_{d \mathop \divides P} d
| r = \paren {\prod_{i \mathop = 0}^{n - 1} 2^i} \paren {\prod_{i \mathop = 0... | Product of Factors of Even Perfect Number | https://proofwiki.org/wiki/Product_of_Factors_of_Even_Perfect_Number | https://proofwiki.org/wiki/Product_of_Factors_of_Even_Perfect_Number | [
"Perfect Numbers"
] | [
"Definition:Perfect Number"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-15261 | Sequence is Bounded in Norm iff Bounded in Metric | Let $\struct {R, \norm {\,\cdot\,} } $ be a normed division ring.
Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a sequence in $R$.
Then:
:$\sequence {x_n} $ is a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,} }$ {{iff}} $\sequence {x_n} $ is ... | === Necessary Condition ===
{{:Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition}}{{qed|lemma}} | 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]] on $R$ be the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|seque... | === [[Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition|Necessary Condition]] ===
{{:Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition}}{{qed|lemma}} | Sequence is Bounded in Norm iff Bounded in Metric | https://proofwiki.org/wiki/Sequence_is_Bounded_in_Norm_iff_Bounded_in_Metric | https://proofwiki.org/wiki/Sequence_is_Bounded_in_Norm_iff_Bounded_in_Metric | [
"Sequence is Bounded in Norm iff Bounded in Metric",
"Normed Division Rings",
"Convergence",
"Metric Spaces"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Sequence",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Normed Division Ring",
"Definition:Bounded Sequence/Metric Space",
"Definition:Metric Space"
... | [
"Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition"
] |
proofwiki-15262 | Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition | Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring.
Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n} $ be a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,}}$
Then:
:$\sequence {x_n} $ is a bou... | Let $\sequence {x_n} $ be a bounded sequence in $\struct {R, \norm {\,\cdot\,} }$.
Then:
:$\exists K \in \R_{\gt 0} : \forall n : \norm {x_n} \le K$
Then $\forall n, m \in \N$:
{{begin-eqn}}
{{eqn | l = \map d { x_n , x_m }
| r = \norm {x_n - x_m}
| c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{... | 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]] on $R$ be the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequen... | Let $\sequence {x_n} $ be a [[Definition:Bounded Sequence in Normed Division Ring|bounded sequence]] in $\struct {R, \norm {\,\cdot\,} }$.
Then:
:$\exists K \in \R_{\gt 0} : \forall n : \norm {x_n} \le K$
Then $\forall n, m \in \N$:
{{begin-eqn}}
{{eqn | l = \map d { x_n , x_m }
| r = \norm {x_n - x_m}
... | Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition | https://proofwiki.org/wiki/Sequence_is_Bounded_in_Norm_iff_Bounded_in_Metric/Necessary_Condition | https://proofwiki.org/wiki/Sequence_is_Bounded_in_Norm_iff_Bounded_in_Metric/Necessary_Condition | [
"Sequence is Bounded in Norm iff Bounded in Metric"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Sequence",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Normed Division Ring",
"Definition:Bounded Sequence/Metric Space",
"Definition:Metric Space"
... | [
"Definition:Bounded Sequence/Normed Division Ring",
"Properties of Norm on Division Ring/Norm of Difference",
"Definition:Sequence",
"Definition:Bounded Sequence/Metric Space",
"Definition:Metric Space"
] |
proofwiki-15263 | Sequence is Bounded in Norm iff Bounded in Metric/Sufficient Condition | Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring.
Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n} $ be a bounded sequence in the metric space $\struct {R, d}$
Then:
:$\sequence {x_n} $ is a bounded sequence in the norm... | Let $\sequence {x_n} $ be a bounded sequence in the metric space $\struct {R, d}$.
Then:
:$\exists K \in \R_{> 0} : \forall n, m : \map d {x_n , x_m} \le K$
By the definition of the metric induced by a norm this is equivalent to:
:$\exists K \in \R_{> 0} : \forall n, m : \norm {x_n - x_m} \le K$
Then $\forall n \in \N$... | 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]] on $R$ be the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequen... | Let $\sequence {x_n} $ be a [[Definition:Bounded Sequence in Metric Space|bounded sequence]] in the [[Definition:Metric Space|metric space]] $\struct {R, d}$.
Then:
:$\exists K \in \R_{> 0} : \forall n, m : \map d {x_n , x_m} \le K$
By the definition of the [[Definition:Metric Induced by Norm on Division Ring|metric ... | Sequence is Bounded in Norm iff Bounded in Metric/Sufficient Condition | https://proofwiki.org/wiki/Sequence_is_Bounded_in_Norm_iff_Bounded_in_Metric/Sufficient_Condition | https://proofwiki.org/wiki/Sequence_is_Bounded_in_Norm_iff_Bounded_in_Metric/Sufficient_Condition | [
"Sequence is Bounded in Norm iff Bounded in Metric"
] | [
"Definition:Normed Division Ring",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Sequence",
"Definition:Bounded Sequence/Metric Space",
"Definition:Metric Space",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Normed Division Ring"
... | [
"Definition:Bounded Sequence/Metric Space",
"Definition:Metric Space",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Sequence",
"Definition:Bounded Sequence/Normed Division Ring",
"Definition:Normed Division Ring"
] |
proofwiki-15264 | Magic Constant of Magic Cube | The magic constant of a magic cube of order $n$ is given by:
:$C_n = \dfrac {n \paren {n^3 + 1} } 2$ | Let $M_n$ denote a magic cube of order $n$.
By Sum of Terms of Magic Cube, the total of all the entries in $M_n$ is given by:
:$T_n = \dfrac {n^3 \paren {n^3 + 1}} 2$
There are $n^2$ rows in $M_n$, each one with the same magic constant.
Thus the magic constant $C_n$ of the magic cube $M_n$ is given by:
{{begin-eqn}}
{{... | The [[Definition:Magic Constant|magic constant]] of a [[Definition:Magic Cube|magic cube]] of [[Definition:Order of Magic Square|order $n$]] is given by:
:$C_n = \dfrac {n \paren {n^3 + 1} } 2$ | Let $M_n$ denote a [[Definition:Magic Cube|magic cube]] of [[Definition:Order of Magic Cube|order $n$]].
By [[Sum of Terms of Magic Cube]], the total of all the entries in $M_n$ is given by:
:$T_n = \dfrac {n^3 \paren {n^3 + 1}} 2$
There are $n^2$ [[Definition:Row of Matrix|rows]] in $M_n$, each one with the same [[... | Magic Constant of Magic Cube | https://proofwiki.org/wiki/Magic_Constant_of_Magic_Cube | https://proofwiki.org/wiki/Magic_Constant_of_Magic_Cube | [
"Magic Cubes"
] | [
"Definition:Magic Square/Magic Constant",
"Definition:Magic Cube",
"Definition:Magic Square/Order"
] | [
"Definition:Magic Cube",
"Definition:Magic Cube/Order",
"Sum of Terms of Magic Cube",
"Definition:Matrix/Row",
"Definition:Magic Square/Magic Constant",
"Definition:Magic Square/Magic Constant",
"Definition:Magic Cube",
"Sum of Terms of Magic Cube"
] |
proofwiki-15265 | Sum of Terms of Magic Cube | The total of all the entries in a magic cube of order $n$ is given by:
:$T_n = \dfrac {n^3 \paren {n^3 + 1} } 2$ | Let $M_n$ denote a magic cube of order $n$.
$M_n$ is by definition an arrangement of the first $n^3$ (strictly) positive integers into an $n \times n \times n$ cubic array containing the positive integers from $1$ upwards.
Thus there are $n^3$ entries in $M_n$, going from $1$ to $n^3$.
Thus:
{{begin-eqn}}
{{eqn | l = T... | The total of all the entries in a [[Definition:Magic Cube|magic cube]] of [[Definition:Order of Magic Cube|order $n$]] is given by:
:$T_n = \dfrac {n^3 \paren {n^3 + 1} } 2$ | Let $M_n$ denote a [[Definition:Magic Cube|magic cube]] of [[Definition:Order of Magic Cube|order $n$]].
$M_n$ is by definition an arrangement of the first $n^3$ [[Definition:Strictly Positive Integer|(strictly) positive integers]] into an $n \times n \times n$ [[Definition:Cube (Geometry)|cubic]] [[Definition:Array|a... | Sum of Terms of Magic Cube | https://proofwiki.org/wiki/Sum_of_Terms_of_Magic_Cube | https://proofwiki.org/wiki/Sum_of_Terms_of_Magic_Cube | [
"Magic Cubes"
] | [
"Definition:Magic Cube",
"Definition:Magic Cube/Order"
] | [
"Definition:Magic Cube",
"Definition:Magic Cube/Order",
"Definition:Strictly Positive/Integer",
"Definition:Cube/Geometry",
"Definition:Array",
"Definition:Positive/Integer",
"Closed Form for Triangular Numbers"
] |
proofwiki-15266 | Smallest Magic Cube is of Order 3 | Apart from the trivial order $1$ magic cube:
{{:Magic Cube/Examples/Order 1}}
the smallest magic cube is the order $3$ magic cube:
{{:Magic Cube/Examples/Order 3}} | Suppose there were an order $2$ magic cube.
Take one row of this magic cube.
From Magic Constant of Magic Cube, the row and column total is $9$.
Any row or column with a $1$ in it must therefore also have an $8$ in it.
But there are:
:one row
:one column
both of which have a $1$ in them.
Therefore the $8$ would need to... | Apart from the trivial [[Magic Cube/Examples/Order 1|order $1$ magic cube]]:
{{:Magic Cube/Examples/Order 1}}
the smallest [[Definition:Magic Cube|magic cube]] is the [[Magic Cube/Examples/Order 3|order $3$ magic cube]]:
{{:Magic Cube/Examples/Order 3}} | Suppose there were an [[Definition:Order of Magic Cube|order $2$]] [[Definition:Magic Cube|magic cube]].
Take one row of this [[Definition:Magic Cube|magic cube]].
From [[Magic Constant of Magic Cube]], the row and column total is $9$.
Any row or column with a $1$ in it must therefore also have an $8$ in it.
But th... | Smallest Magic Cube is of Order 3 | https://proofwiki.org/wiki/Smallest_Magic_Cube_is_of_Order_3 | https://proofwiki.org/wiki/Smallest_Magic_Cube_is_of_Order_3 | [
"Magic Cubes"
] | [
"Magic Cube/Examples/Order 1",
"Definition:Magic Cube",
"Magic Cube/Examples/Order 3"
] | [
"Definition:Magic Cube/Order",
"Definition:Magic Cube",
"Definition:Magic Cube",
"Magic Constant of Magic Cube",
"Definition:Magic Cube",
"Definition:Magic Cube/Order",
"Definition:Magic Cube",
"Category:Magic Cubes"
] |
proofwiki-15267 | Fourth Power as Summation of Groups of Consecutive Integers | Take the positive integers and group them in sets such that the $m$th set contains the next $m$ positive integers:
:$\set 1, \set {2, 3}, \set {4, 5, 6}, \set {7, 8, 9, 10}, \set {11, 12, 13, 14, 15}, \ldots$
Remove all the sets with an even number of elements.
Then the sum of all the integers in the first $n$ sets rem... | Let $S_m$ be the $m$th set of $m$ consecutive integers.
Let $S_k$ be the $k$th set of $m$ consecutive integers after the sets with an even number of elements have been removed.
Then $S_k = S_m$ where $m = 2 k - 1$.
By the method of construction:
:the largest integer in $S_m$ is $T_m$, the $m$th triangular number
:there... | Take the [[Definition:Strictly Positive Integer|positive integers]] and group them in [[Definition:Set|sets]] such that the $m$th [[Definition:Set|set]] contains the next $m$ [[Definition:Strictly Positive Integer|positive integers]]:
:$\set 1, \set {2, 3}, \set {4, 5, 6}, \set {7, 8, 9, 10}, \set {11, 12, 13, 14, 15},... | Let $S_m$ be the $m$th [[Definition:Set|set]] of $m$ consecutive [[Definition:Integer|integers]].
Let $S_k$ be the $k$th [[Definition:Set|set]] of $m$ consecutive [[Definition:Integer|integers]] after the [[Definition:Set|sets]] with an [[Definition:Even Integer|even number]] of [[Definition:Element|elements]] have be... | Fourth Power as Summation of Groups of Consecutive Integers | https://proofwiki.org/wiki/Fourth_Power_as_Summation_of_Groups_of_Consecutive_Integers | https://proofwiki.org/wiki/Fourth_Power_as_Summation_of_Groups_of_Consecutive_Integers | [
"Fourth Powers",
"Fourth Power as Summation of Groups of Consecutive Integers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Even Integer",
"Definition:Element",
"Definition:Addition/Integers",
"Definition:Integer",
"Definition:Set"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Set",
"Definition:Integer",
"Definition:Set",
"Definition:Even Integer",
"Definition:Element",
"Definition:Integer",
"Definition:Triangular Number",
"Definition:Integer",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Ele... |
proofwiki-15268 | Even Power of 3 as Sum of Consecutive Positive Integers | Take the positive integers and group them in sets such that the $n$th set contains the next $3^n$ positive integers:
:$\set 1, \set {2, 3, 4}, \set {5, 6, \ldots, 13}, \set {14, 15, \cdots, 40}, \ldots$
Let the $n$th such set be denoted $S_{n - 1}$, that is, letting $S_0 := \set 1$ be considered as the zeroth.
Then the... | The total number of elements in $S_0, S_1, \ldots, S_r$ is:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^r \card {S_j}
| r = \sum_{j \mathop = 0}^r 3^j
| c =
}}
{{eqn | r = \dfrac {3^{r + 1} - 1} {3 - 1}
| c = Sum of Geometric Sequence
}}
{{eqn | r = \dfrac {3^{r + 1} - 1} 2
| c = simplifying... | Take the [[Definition:Strictly Positive Integer|positive integers]] and group them in [[Definition:Set|sets]] such that the $n$th [[Definition:Set|set]] contains the next $3^n$ [[Definition:Strictly Positive Integer|positive integers]]:
:$\set 1, \set {2, 3, 4}, \set {5, 6, \ldots, 13}, \set {14, 15, \cdots, 40}, \ldot... | The total number of [[Definition:Element|elements]] in $S_0, S_1, \ldots, S_r$ is:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^r \card {S_j}
| r = \sum_{j \mathop = 0}^r 3^j
| c =
}}
{{eqn | r = \dfrac {3^{r + 1} - 1} {3 - 1}
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | r = \dfrac {3^{r + 1} - ... | Even Power of 3 as Sum of Consecutive Positive Integers | https://proofwiki.org/wiki/Even_Power_of_3_as_Sum_of_Consecutive_Positive_Integers | https://proofwiki.org/wiki/Even_Power_of_3_as_Sum_of_Consecutive_Positive_Integers | [
"Powers of 3",
"Even Power of 3 as Sum of Consecutive Positive Integers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Zeroth",
"Definition:Addition/Integers",
"Definition:Element"
] | [
"Definition:Element",
"Sum of Geometric Sequence",
"Definition:Iverson's Convention",
"Definition:Subtraction/Integers",
"Definition:Triangular Number",
"Closed Form for Triangular Numbers",
"Difference of Two Squares"
] |
proofwiki-15269 | Convergent Subsequence of Cauchy Sequence/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $\sequence{x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {R, \norm {\,\cdot\,} }$.
Let $x \in R$.
Then $\sequence {x_n}$ converges to $x$ {{iff}} $\sequence {x_n}$ has a subsequence that converges to $x$. | Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
By the definition of a convergent sequence in a normed division ring then:
:$\sequence {x_n}$ converges to $x$ in $\struct {R, \norm {\,\cdot\,} }$ {{iff}} $\sequence {x_n}$ converges to $x$ in $\struct {R, d}$.
By Convergent Subsequence of Cauchy Se... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence{x_n}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence (Normed Division Ring)|Cauchy sequence]] in $\struct {R, \norm {\,\cdot\,} }$.
Let $x \in R$.
Then $\sequence {x_n}$ [[Definition:Convergent... | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] on $R$ be the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
By the definition of a [[Definition:Convergent Sequence in Normed Division Ring|convergent sequence in a normed division ring]] then:
:$\sequence {x_n}$ [[D... | Convergent Subsequence of Cauchy Sequence/Normed Division Ring | https://proofwiki.org/wiki/Convergent_Subsequence_of_Cauchy_Sequence/Normed_Division_Ring | https://proofwiki.org/wiki/Convergent_Subsequence_of_Cauchy_Sequence/Normed_Division_Ring | [
"Convergent Sequences in Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Metric Space",
"Convergent Subsequence of Cauchy Sequence/Metric Space",
"D... |
proofwiki-15270 | Null Sequences form Maximal Left and Right Ideal | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\CC$ be the ring of Cauchy sequences over $R$.
Let $\NN$ be the set of null sequences.
That is:
:$\ds \NN = \set {\sequence {x_n}: \lim_{n \mathop \to \infty} x_n = 0 }$
Then $\NN$ is a ring ideal of $\CC$ that is a maximal left ideal and a maximal... | By every convergent sequence is a Cauchy sequence then $\NN \subseteq \CC$.
The proof is completed in these steps:
:$(1): \quad \NN$ is an ideal of $\CC$.
:$(2): \quad \NN$ is a maximal left ideal.
:$(3): \quad \NN$ is a maximal right ideal. | Let $\struct {R, \norm {\, \cdot \,} }$ 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 [[Definition:Set|set]] of [[Definition:Null Sequence in Normed Division Ring|null sequences]].
That is:
:... | By [[Convergent Sequence is Cauchy Sequence/Normed Division Ring|every convergent sequence is a Cauchy sequence]] then $\NN \subseteq \CC$.
The proof is completed in these steps:
:$(1): \quad \NN$ is an [[Definition:Ideal of Ring|ideal]] of $\CC$.
:$(2): \quad \NN$ is a [[Definition:Maximal Left Ideal of Ring|maximal... | Null Sequences form Maximal Left and Right Ideal | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal | [
"Cauchy Sequences",
"Normed Division Rings",
"Null Sequences form Maximal Left and Right Ideal"
] | [
"Definition:Normed Division Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Ideal of Ring",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Maximal Ideal of Ring/Right"
] | [
"Convergent Sequence is Cauchy Sequence/Normed Division Ring",
"Definition:Ideal of Ring",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Maximal Ideal of Ring/Right"
] |
proofwiki-15271 | Null Sequences form Maximal Left and Right Ideal/Lemma 1 | :$\NN$ is an ideal of $\CC$. | The Test for Ideal is applied to prove the result. | :$\NN$ is an [[Definition:Ideal of Ring|ideal]] of $\CC$. | The [[Test for Ideal]] is applied to prove the result. | Null Sequences form Maximal Left and Right Ideal/Lemma 1 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_1 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_1 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [
"Definition:Ideal of Ring"
] | [
"Test for Ideal",
"Test for Ideal"
] |
proofwiki-15272 | Null Sequences form Maximal Left and Right Ideal/Lemma 2 | :$\NN$ is a maximal left ideal. | By Lemma 1 of Null Sequences form Maximal Left and Right Ideal:
:$\NN$ is an ideal of $\CC$.
Hence $\NN$ is a left ideal of $\CC$.
It remains to show that $\NN$ is maximal. | :$\NN$ is a [[Definition:Maximal Left Ideal of Ring|maximal left ideal]]. | By [[Null Sequences form Maximal Left and Right Ideal/Lemma 1|Lemma 1 of Null Sequences form Maximal Left and Right Ideal]]:
:$\NN$ is an [[Definition:Ideal of Ring|ideal]] of $\CC$.
Hence $\NN$ is a [[Definition:Left Ideal of Ring|left ideal]] of $\CC$.
It remains to show that $\NN$ is [[Definition:Maximal Left Idea... | Null Sequences form Maximal Left and Right Ideal/Lemma 2 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_2 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_2 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [
"Definition:Maximal Ideal of Ring/Left"
] | [
"Null Sequences form Maximal Left and Right Ideal/Lemma 1",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Maximal Ideal of Ring/Left"
] |
proofwiki-15273 | Null Sequences form Maximal Left and Right Ideal/Lemma 3 | :$\NN$ is a maximal right ideal. | By Lemma 1 of Null Sequences form Maximal Left and Right Ideal then $\NN$ is an ideal of $\CC$.
Hence $\NN$ is a right ideal of $\CC$.
It remains to show that $\NN$ is maximal.
By Lemma 7 of Null Sequences form Maximal Left and Right Ideal then $\NN \subsetneq \CC$.
By maximal right ideal it needs to be shown that:
:Th... | :$\NN$ is a [[Definition:Maximal Right Ideal of Ring|maximal right ideal]]. | By [[Null Sequences form Maximal Left and Right Ideal/Lemma 1|Lemma 1 of Null Sequences form Maximal Left and Right Ideal]] then $\NN$ is an [[Definition:Ideal of Ring|ideal]] of $\CC$.
Hence $\NN$ is a [[Definition:Right Ideal of Ring|right ideal]] of $\CC$.
It remains to show that $\NN$ is [[Definition:Maximal Righ... | Null Sequences form Maximal Left and Right Ideal/Lemma 3 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_3 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_3 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [
"Definition:Maximal Ideal of Ring/Right"
] | [
"Null Sequences form Maximal Left and Right Ideal/Lemma 1",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring/Right Ideal",
"Definition:Maximal Ideal of Ring/Right",
"Null Sequences form Maximal Left and Right Ideal/Lemma 7",
"Definition:Maximal Ideal of Ring/Right",
"Definition:Ideal of Ring/Right ... |
proofwiki-15274 | Product of Sequence of Fermat Numbers plus 2 | Let $F_n$ denote the $n$th Fermat number.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_n
| r = \prod_{j \mathop = 0}^{n - 1} F_j + 2
| c =
}}
{{eqn | r = F_0 F_1 \dotsm F_{n - 1} + 2
| c =
}}
{{end-eqn}} | The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$F_n = \ds \prod_{j \mathop = 0}^{n - 1} F_j + 2$ | Let $F_n$ denote the $n$th [[Definition:Fermat Number|Fermat number]].
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_n
| r = \prod_{j \mathop = 0}^{n - 1} F_j + 2
| c =
}}
{{eqn | r = F_0 F_1 \dotsm F_{n - 1} + 2
| c =
}}
{{end-eqn}} | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$F_n = \ds \prod_{j \mathop = 0}^{n - 1} F_j + 2$ | Product of Sequence of Fermat Numbers plus 2 | https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 | https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 | [
"Fermat Numbers"
] | [
"Definition:Fermat Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-15275 | 492 is Sum of 3 Cubes in 3 Ways | $492$ can be expressed as the sum of $3$ cubes, either positive or negative in $3$ known ways.
{{begin-eqn}}
{{eqn | l = 492
| r = 50^3 + \paren {-19}^3 + \paren {-49}^3
}}
{{eqn | r = 123 \, 134^3 + 9179^3 + \paren {-123 \, 151}^3
}}
{{eqn | r = 1 \, 793 \, 337 \, 644^3 + \paren {-81 \, 3701 \, 167}^3 + \paren {... | Brute force. | $492$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Cube Number|cubes]], either [[Definition:Positive Integer|positive]] or [[Definition:Negative Integer|negative]] in $3$ known ways.
{{begin-eqn}}
{{eqn | l = 492
| r = 50^3 + \paren {-19}^3 + \paren {-49}^3
}}
{{eqn | r = 123 \... | Brute force. | 492 is Sum of 3 Cubes in 3 Ways | https://proofwiki.org/wiki/492_is_Sum_of_3_Cubes_in_3_Ways | https://proofwiki.org/wiki/492_is_Sum_of_3_Cubes_in_3_Ways | [
"492",
"Sums of Cubes"
] | [
"Definition:Addition/Integers",
"Definition:Cube Number",
"Definition:Positive/Integer",
"Definition:Negative/Integer"
] | [] |
proofwiki-15276 | Smallest n needing 6 Numbers less than n so that Product of Factorials is Square | Let $n \in \Z_{>0}$ be a positive integer.
Then it is possible to choose at most $6$ positive integers less than $n$ such that the product of their factorials is square.
The smallest $n$ that actually requires $6$ numbers to be chosen is $527$. | Obviously the product cannot be a square if $n$ is a prime.
For $n$ composite, we can write:
:$n = a b$
where $a, b \in \Z_{>1}$.
Then:
{{begin-eqn}}
{{eqn | o =
| r = n! \paren {n - 1}! \paren {a!} \paren {a - 1}! \paren {b!} \paren {b - 1}!
}}
{{eqn | r = n a b \paren {\paren {n - 1}! \paren {a - 1}! \paren {b ... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Then it is possible to choose at most $6$ [[Definition:Positive Integer|positive integers]] less than $n$ such that the [[Definition:Integer Multiplication|product]] of their [[Definition:Factorial|factorials]] is [[Definition:Square Number|squ... | Obviously the product cannot be a [[Definition:Square Number|square]] if $n$ is a [[Definition:Prime Number|prime]].
For $n$ [[Definition:Composite Number|composite]], we can write:
:$n = a b$
where $a, b \in \Z_{>1}$.
Then:
{{begin-eqn}}
{{eqn | o =
| r = n! \paren {n - 1}! \paren {a!} \paren {a - 1}! \paren ... | Smallest n needing 6 Numbers less than n so that Product of Factorials is Square | https://proofwiki.org/wiki/Smallest_n_needing_6_Numbers_less_than_n_so_that_Product_of_Factorials_is_Square | https://proofwiki.org/wiki/Smallest_n_needing_6_Numbers_less_than_n_so_that_Product_of_Factorials_is_Square | [
"527",
"Factorials",
"Square Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Multiplication/Integers",
"Definition:Factorial",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Square Number",
"Definition:Factorial",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square-Free Integer",
"Definition:Square-Free Integer",
"Defin... |
proofwiki-15277 | Null Sequences form Maximal Left and Right Ideal/Lemma 4 | :$\NN \ne \O$ | From Constant Sequence Converges to Constant in Normed Division Ring, the zero $\tuple {0, 0, 0, \dots}$ of $\CC$ to converges $0 \in R$.
Therefore $\tuple {0, 0, 0, \dots} \in \NN$.
{{qed}} | :$\NN \ne \O$ | From [[Constant Sequence Converges to Constant in Normed Division Ring]], the [[Definition:Ring Zero|zero]] $\tuple {0, 0, 0, \dots}$ of $\CC$ to [[Definition:Convergent Sequence in Normed Division Ring|converges]] $0 \in R$.
Therefore $\tuple {0, 0, 0, \dots} \in \NN$.
{{qed}} | Null Sequences form Maximal Left and Right Ideal/Lemma 4 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_4 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_4 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [] | [
"Constant Sequence Converges to Constant in Normed Division Ring",
"Definition:Ring Zero",
"Definition:Convergent Sequence/Normed Division Ring"
] |
proofwiki-15278 | Null Sequences form Maximal Left and Right Ideal/Lemma 5 | :$\forall \sequence {x_n}, \sequence {y_n} \in \NN: \sequence {x_n} + \paren {-\sequence {y_n} } \in \NN$ | Let $\ds \lim_{n \mathop \to \infty} x_n = 0$ and $\ds \lim_{n \mathop \to \infty} y_n = 0$.
The sequence $\sequence {x_n} + \paren {-\sequence {y_n} } = \sequence {x_n - y_n}$.
From Difference Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} x_n - y_n = 0 - 0 = 0$
The result follows.
{{qed... | :$\forall \sequence {x_n}, \sequence {y_n} \in \NN: \sequence {x_n} + \paren {-\sequence {y_n} } \in \NN$ | Let $\ds \lim_{n \mathop \to \infty} x_n = 0$ and $\ds \lim_{n \mathop \to \infty} y_n = 0$.
The [[Definition:Sequence|sequence]] $\sequence {x_n} + \paren {-\sequence {y_n} } = \sequence {x_n - y_n}$.
From [[Difference Rule for Sequences in Normed Division Ring]]:
:$\ds \lim_{n \mathop \to \infty} x_n - y_n = 0 - 0 ... | Null Sequences form Maximal Left and Right Ideal/Lemma 5 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_5 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_5 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [] | [
"Definition:Sequence",
"Combination Theorem for Sequences/Normed Division Ring/Difference Rule"
] |
proofwiki-15279 | Null Sequences form Maximal Left and Right Ideal/Lemma 6 | :$\quad \forall \sequence {x_n} \in \NN, \sequence {y_n} \in \CC: \sequence {x_n} \sequence {y_n} \in \NN, \sequence {y_n} \sequence {x_n} \in \NN$ | Let $\ds \lim_{n \mathop \to \infty} x_n = 0$.
By the definition of the product on the ring of Cauchy sequences then:
{{begin-eqn}}
{{eqn | l = \sequence {x_n} \sequence {y_n}
| r = \sequence {x_n y_n}
}}
{{eqn | l = \sequence {y_n} \sequence {x_n}
| r = \sequence {y_n x_n}
}}
{{end-eqn}}
By Product of Sequ... | :$\quad \forall \sequence {x_n} \in \NN, \sequence {y_n} \in \CC: \sequence {x_n} \sequence {y_n} \in \NN, \sequence {y_n} \sequence {x_n} \in \NN$ | Let $\ds \lim_{n \mathop \to \infty} x_n = 0$.
By the definition of the [[Definition:Ring of Cauchy Sequences|product on the ring of Cauchy sequences]] then:
{{begin-eqn}}
{{eqn | l = \sequence {x_n} \sequence {y_n}
| r = \sequence {x_n y_n}
}}
{{eqn | l = \sequence {y_n} \sequence {x_n}
| r = \sequence ... | Null Sequences form Maximal Left and Right Ideal/Lemma 6 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_6 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_6 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [] | [
"Definition:Ring of Cauchy Sequences",
"Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero"
] |
proofwiki-15280 | Null Sequences form Maximal Left and Right Ideal/Lemma 7 | :$\NN \subsetneq \CC$. | By every convergent sequence is a Cauchy sequence then $\NN \subseteq \CC$.
From Constant Sequence Converges to Constant in Normed Division Ring, the unity $\tuple {1, 1, 1, \dotsc}$ of $\CC$ converges to $1 \in R$, and therefore $\tuple {1, 1, 1, \dotsc} \in \CC \setminus \NN$
So $\NN \subsetneq \CC$.
{{qed}} | :$\NN \subsetneq \CC$. | By [[Convergent Sequence is Cauchy Sequence/Normed Division Ring|every convergent sequence is a Cauchy sequence]] then $\NN \subseteq \CC$.
From [[Constant Sequence Converges to Constant in Normed Division Ring]], the [[Definition:Unity of Ring|unity]] $\tuple {1, 1, 1, \dotsc}$ of $\CC$ [[Definition:Convergent Sequen... | Null Sequences form Maximal Left and Right Ideal/Lemma 7 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_7 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_7 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [] | [
"Convergent Sequence is Cauchy Sequence/Normed Division Ring",
"Constant Sequence Converges to Constant in Normed Division Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] |
proofwiki-15281 | Null Sequences form Maximal Left and Right Ideal/Lemma 8 | :There is no left ideal $\JJ$ of $\CC$ such that $\NN \subsetneq \JJ \subsetneq \CC$ | Let $\JJ$ be a left ideal of $\CC$ such that $\NN \subsetneq \JJ \subseteq \CC$.
It will be shown that $\JJ$ = $\CC$, from which the result will follow.
Let $\sequence {x_n} \in \JJ \setminus \NN$
By Inverse Rule for Cauchy sequences then
:$\exists K \in \N: \sequence { \paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ i... | :There is no [[Definition:Left Ideal of Ring|left ideal]] $\JJ$ of $\CC$ such that $\NN \subsetneq \JJ \subsetneq \CC$ | Let $\JJ$ be a [[Definition:Left Ideal of Ring|left ideal]] of $\CC$ such that $\NN \subsetneq \JJ \subseteq \CC$.
It will be shown that $\JJ$ = $\CC$, from which the result will follow.
Let $\sequence {x_n} \in \JJ \setminus \NN$
By [[Combination Theorem for Cauchy Sequences/Inverse Rule|Inverse Rule for Cauchy seq... | Null Sequences form Maximal Left and Right Ideal/Lemma 8 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_8 | https://proofwiki.org/wiki/Null_Sequences_form_Maximal_Left_and_Right_Ideal/Lemma_8 | [
"Null Sequences form Maximal Left and Right Ideal"
] | [
"Definition:Ideal of Ring/Left Ideal"
] | [
"Definition:Ideal of Ring/Left Ideal",
"Combination Theorem for Cauchy Sequences/Inverse Rule",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Sequence",
"Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Unity (Abstrac... |
proofwiki-15282 | Difference between Two Squares equal to Repdigit | {{begin-eqn}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 56^2 - 45^2
| r = 1111
| c =
}}
{{eqn | l = 556^2 - 445^2
| r = 111 \, 111
| c =
}}
{{eqn | o = :
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 7^2 - 4^2
| r = 33
| c =
}}
{{eqn | l = 67^2 - 34^... | Let $a, b$ be integers with $1 \le b < a \le 8$ and $a + b = 9$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + \sum_{k \mathop = 0}^n a 10^k}^2 - \paren {1 + \sum_{k \mathop = 0}^n b 10^k}^2
| r = \paren {1 + \sum_{k \mathop = 0}^n a 10^k - 1 - \sum_{k \mathop = 0}^n b 10^k} \paren {1 + \sum_{k \mathop = 0}^n a 10^... | {{begin-eqn}}
{{eqn | l = 6^2 - 5^2
| r = 11
| c =
}}
{{eqn | l = 56^2 - 45^2
| r = 1111
| c =
}}
{{eqn | l = 556^2 - 445^2
| r = 111 \, 111
| c =
}}
{{eqn | o = :
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 7^2 - 4^2
| r = 33
| c =
}}
{{eqn | l = 67^2 - 3... | Let $a, b$ be [[Definition:Integer|integers]] with $1 \le b < a \le 8$ and $a + b = 9$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + \sum_{k \mathop = 0}^n a 10^k}^2 - \paren {1 + \sum_{k \mathop = 0}^n b 10^k}^2
| r = \paren {1 + \sum_{k \mathop = 0}^n a 10^k - 1 - \sum_{k \mathop = 0}^n b 10^k} \paren {1 + \su... | Difference between Two Squares equal to Repdigit | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repdigit | https://proofwiki.org/wiki/Difference_between_Two_Squares_equal_to_Repdigit | [
"Difference between Two Squares equal to Repunit"
] | [] | [
"Definition:Integer",
"Difference of Two Squares",
"Difference of Two Squares",
"Translation of Index Variable of Summation",
"Definition:Repdigit Number"
] |
proofwiki-15283 | Largest Number Not Expressible as Sum of Fewer than 8 Cubes | $8042$ is (probably) the largest positive integer that cannot be expressed as the sum of fewer than $8$ cubes. | It is believed that this entry is a mistake.
$8042 = 1^3 + 4^3 + 4^3 + 10^3 + 10^3 + 10^3 + 17^3$, among many other expressions.
However:
$8042$ is conjectured to be the largest positive integer that cannot be expressed as the sum of fewer than $\bf 7$ cubes.
$\bf {454}$ is proven to be the largest positive integer tha... | $8042$ is (probably) the largest [[Definition:Positive Integer|positive integer]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of fewer than $8$ [[Definition:Cube Number|cubes]]. | It is believed that this entry is a mistake.
$8042 = 1^3 + 4^3 + 4^3 + 10^3 + 10^3 + 10^3 + 17^3$, among many other expressions.
However:
$8042$ is conjectured to be the largest [[Definition:Positive Integer|positive integer]] that cannot be expressed as the [[Definition:Integer Addition|sum]] of fewer than $\bf 7$ ... | Largest Number Not Expressible as Sum of Fewer than 8 Cubes | https://proofwiki.org/wiki/Largest_Number_Not_Expressible_as_Sum_of_Fewer_than_8_Cubes | https://proofwiki.org/wiki/Largest_Number_Not_Expressible_as_Sum_of_Fewer_than_8_Cubes | [
"Sums of Cubes",
"8042"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Cube Number"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Cube Number",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Cube Number"
] |
proofwiki-15284 | Infinite Number of Even Fermat Pseudoprimes | Despite their relative rarity, there exist an infinite number of even Fermat pseudoprimes. | In the context of Wells, Fermat pseudoprime probably means Fermat pseudoprimes to the base $2$.
Consider the equation:
:$2^m - 2 \equiv 0 \pmod m$
Any $m$ satisfying the above equation is a Fermat pseudoprime.
We show that for each even $m$ satisfying the above equation, there exists a prime $p$ such that $m p$ also sa... | Despite their relative rarity, there exist an [[Definition:Infinite Set|infinite number]] of [[Definition:Even Integer|even]] [[Definition:Fermat Pseudoprime|Fermat pseudoprimes]]. | In the context of Wells, [[Definition:Fermat Pseudoprime|Fermat pseudoprime]] probably means [[Definition:Poulet Number|Fermat pseudoprimes to the base $2$]].
Consider the equation:
:$2^m - 2 \equiv 0 \pmod m$
Any $m$ satisfying the above equation is a [[Definition:Fermat Pseudoprime|Fermat pseudoprime]].
We show t... | Infinite Number of Even Fermat Pseudoprimes | https://proofwiki.org/wiki/Infinite_Number_of_Even_Fermat_Pseudoprimes | https://proofwiki.org/wiki/Infinite_Number_of_Even_Fermat_Pseudoprimes | [
"Fermat Pseudoprimes"
] | [
"Definition:Infinite Set",
"Definition:Even Integer",
"Definition:Fermat Pseudoprime"
] | [
"Definition:Fermat Pseudoprime",
"Definition:Poulet Number",
"Definition:Fermat Pseudoprime",
"Definition:Prime Number",
"Zsigmondy's Theorem",
"Definition:Prime Number",
"Fermat's Little Theorem",
"Chinese Remainder Theorem",
"Congruence by Product of Moduli",
"Definition:Even Integer",
"Defini... |
proofwiki-15285 | Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit | The following sequence of $5$ consecutive positive integers cannot be made into prime numbers by changing just one digit:
:$872\,894, 872\,895, 872\,896, 872\,897, 872\,898$
{{OEIS|A192545}} | Numbers ending in $0$, $2$, $4$, $6$ and $8$ are not prime because by Divisibility by 2 they are divisible by $2$.
Numbers ending in $0$ and $5$ are not prime because by Divisibility by 5 they are divisible by $5$.
Hence each of $872\,894$, $872\,895$, $872\,896$ and $872\,898$ remain composite when you change any of t... | The following [[Definition:Integer Sequence|sequence]] of $5$ consecutive [[Definition:Positive Integer|positive integers]] cannot be made into [[Definition:Prime Number|prime numbers]] by changing just one [[Definition:Digit|digit]]:
:$872\,894, 872\,895, 872\,896, 872\,897, 872\,898$
{{OEIS|A192545}} | Numbers ending in $0$, $2$, $4$, $6$ and $8$ are not [[Definition:Prime Number|prime]] because by [[Divisibility by 2]] they are [[Definition:Divisor of Integer|divisible]] by $2$.
Numbers ending in $0$ and $5$ are not [[Definition:Prime Number|prime]] because by [[Divisibility by 5]] they are [[Definition:Divisor of ... | Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit | https://proofwiki.org/wiki/Sequence_of_5_Consecutive_Non-Primable_Numbers_by_Changing_1_Digit | https://proofwiki.org/wiki/Sequence_of_5_Consecutive_Non-Primable_Numbers_by_Changing_1_Digit | [
"Prime Numbers",
"Recreational Mathematics",
"Numbers that cannot be made Prime by changing 1 Digit"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Prime Number",
"Definition:Digit"
] | [
"Definition:Prime Number",
"Divisibility by 2",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Divisibility by 5",
"Definition:Divisor (Algebra)/Integer",
"Definition:Composite Number",
"Definition:Prime Factor",
"Definition:Prime Number",
"Definition:Prime Number",
"Definit... |
proofwiki-15286 | Smith Numbers are Infinite in Number | There are infinitely many Smith numbers. | First we prove the the algorithm above does generate Smith numbers.
Let $n \ge 2$.
We have:
:$m = 10^n - 1 = 3 \times 3 \times R_n$
where $R_n$ is the repunit with $n$ digits.
We apply the Lemma, taking note that $r \ge 3$:
:$\map {S_p} m < 9 \map N m - 0.54 \times 3 = 9 n - 1.62$
Since both $\map {S_p} m$ and $9 n$ ar... | There are [[Definition:Infinitely Many|infinitely many]] [[Definition:Smith Number|Smith numbers]]. | First we prove the the algorithm above does generate [[Definition:Smith Number|Smith numbers]].
Let $n \ge 2$.
We have:
:$m = 10^n - 1 = 3 \times 3 \times R_n$
where $R_n$ is the [[Definition:Repunit|repunit]] with $n$ [[Definition:Digit|digits]].
We apply the [[Smith Numbers are Infinite in Number/Lemma|Lemma]], t... | Smith Numbers are Infinite in Number | https://proofwiki.org/wiki/Smith_Numbers_are_Infinite_in_Number | https://proofwiki.org/wiki/Smith_Numbers_are_Infinite_in_Number | [
"Smith Numbers",
"Smith Numbers are Infinite in Number"
] | [
"Definition:Infinite Set",
"Definition:Smith Number",
"Definition:Smith Number"
] | [
"Definition:Smith Number",
"Definition:Repunit",
"Definition:Digit",
"Smith Numbers are Infinite in Number/Lemma",
"Definition:Integer",
"Division Theorem",
"Definition:Integer",
"Definition:Smith Number",
"Multiple of Repdigit Base minus 1/Generalization",
"Multiple of Repdigit Base minus 1/Gener... |
proofwiki-15287 | Construction of Smith Number from Prime Repunit | Let $R_n$ be a repunit which is prime where $n \ge 3$.
Then $3304 \times R_n$ is a Smith number.
$3304$ is not the only number this works for, neither is it the smallest, but it serves as an example of the technique. | Let $\map S n$ denote the sum of the digits of a positive integer $n$.
Let $\map {S_p} n$ denote the sum of the digits of the prime decomposition of $n$.
Then $\map S n = \map {S_p} n$ {{iff}} $n$ is a Smith number.
Let $n \ge 3$.
We have that:
:$3304 = 2 \times 2 \times 2 \times 7 \times 59$
and so for a prime repunit... | Let $R_n$ be a [[Definition:Repunit|repunit]] which is [[Definition:Prime Number|prime]] where $n \ge 3$.
Then $3304 \times R_n$ is a [[Definition:Smith Number|Smith number]].
$3304$ is not the only number this works for, neither is it the smallest, but it serves as an example of the technique. | Let $\map S n$ denote the [[Definition:Integer Addition|sum]] of the [[Definition:Digit|digits]] of a [[Definition:Positive Integer|positive integer]] $n$.
Let $\map {S_p} n$ denote the [[Definition:Integer Addition|sum]] of the [[Definition:Digit|digits]] of the [[Definition:Prime Decomposition|prime decomposition]] ... | Construction of Smith Number from Prime Repunit | https://proofwiki.org/wiki/Construction_of_Smith_Number_from_Prime_Repunit | https://proofwiki.org/wiki/Construction_of_Smith_Number_from_Prime_Repunit | [
"Smith Numbers",
"Repunits"
] | [
"Definition:Repunit",
"Definition:Prime Number",
"Definition:Smith Number"
] | [
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Prime Decomposition",
"Definition:Smith Number",
"Definition:Prime Number",
"Definition:Repunit",
"Basis Representation Theorem",
"Basis Representati... |
proofwiki-15288 | Arithmetic Sequence of 16 Primes | The $16$ integers in arithmetic sequence defined as:
:$2\,236\,133\,941 + 223\,092\,870 n$
are prime for $n = 0, 1, \ldots, 15$. | First we note that:
:$2\,236\,133\,941 - 223\,092\,870 = 2\,013\,041\,071 = 53 \times 89 \times 426\,763$
and so this arithmetic sequence of primes does not extend to $n < 0$.
{{begin-eqn}}
{{eqn | l = 2\,236\,133\,941 + 0 \times 223\,092\,870
| r = 2\,236\,133\,941
| c = which is prime
}}
{{eqn | l = 2\,23... | The $16$ [[Definition:Integer|integers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] defined as:
:$2\,236\,133\,941 + 223\,092\,870 n$
are [[Definition:Prime Number|prime]] for $n = 0, 1, \ldots, 15$. | First we note that:
:$2\,236\,133\,941 - 223\,092\,870 = 2\,013\,041\,071 = 53 \times 89 \times 426\,763$
and so this [[Definition:Arithmetic Sequence|arithmetic sequence]] of [[Definition:Prime Number|primes]] does not extend to $n < 0$.
{{begin-eqn}}
{{eqn | l = 2\,236\,133\,941 + 0 \times 223\,092\,870
| r... | Arithmetic Sequence of 16 Primes | https://proofwiki.org/wiki/Arithmetic_Sequence_of_16_Primes | https://proofwiki.org/wiki/Arithmetic_Sequence_of_16_Primes | [
"Prime Numbers",
"Arithmetic Sequences"
] | [
"Definition:Integer",
"Definition:Arithmetic Sequence",
"Definition:Prime Number"
] | [
"Definition:Arithmetic Sequence",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime N... |
proofwiki-15289 | Smallest 17 Primes in Arithmetic Sequence | The smallest $17$ primes in arithmetic sequence are:
:$3\,430\,751\,869 + 87\,297\,210 n$
for $n = 0, 1, \ldots, 16$. | First we note that:
:$3\,430\,751\,869 - 87\,297\,210 = 3\,343\,454\,659 = 17\,203 \times 194\,353$
and so this arithmetic sequence of primes does not extend to $n < 0$.
{{begin-eqn}}
{{eqn | l = 3\,430\,751\,869 + 0 \times 87\,297\,210
| r = 3\,430\,751\,869
| c = which is prime
}}
{{eqn | l = 3\,430\,751\... | The smallest $17$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
:$3\,430\,751\,869 + 87\,297\,210 n$
for $n = 0, 1, \ldots, 16$. | First we note that:
:$3\,430\,751\,869 - 87\,297\,210 = 3\,343\,454\,659 = 17\,203 \times 194\,353$
and so this [[Definition:Arithmetic Sequence|arithmetic sequence]] of [[Definition:Prime Number|primes]] does not extend to $n < 0$.
{{begin-eqn}}
{{eqn | l = 3\,430\,751\,869 + 0 \times 87\,297\,210
| r = 3\,4... | Smallest 17 Primes in Arithmetic Sequence | https://proofwiki.org/wiki/Smallest_17_Primes_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Smallest_17_Primes_in_Arithmetic_Sequence | [
"Prime Numbers",
"Arithmetic Sequences"
] | [
"Definition:Prime Number",
"Definition:Arithmetic Sequence"
] | [
"Definition:Arithmetic Sequence",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime N... |
proofwiki-15290 | Smallest 18 Primes in Arithmetic Sequence | The smallest $18$ primes in arithmetic sequence are:
:$107\,928\,278\,317 + 9\,922\,782\,870 n$
for $n = 0, 1, \ldots, 16$. | First we note that:
:$107\,928\,278\,317 - 9\,922\,782\,870 = 98\,005\,495\,447 = 29 \times 149 \times 22\,681\,207$
and so this arithmetic sequence of primes does not extend to $n < 0$.
{{begin-eqn}}
{{eqn | l = 107\,928\,278\,317 + 0 \times 9\,922\,782\,870
| r = 107\,928\,278\,317
| c = which is prime
}}... | The smallest $18$ [[Definition:Prime Number|primes]] in [[Definition:Arithmetic Sequence|arithmetic sequence]] are:
:$107\,928\,278\,317 + 9\,922\,782\,870 n$
for $n = 0, 1, \ldots, 16$. | First we note that:
:$107\,928\,278\,317 - 9\,922\,782\,870 = 98\,005\,495\,447 = 29 \times 149 \times 22\,681\,207$
and so this [[Definition:Arithmetic Sequence|arithmetic sequence]] of [[Definition:Prime Number|primes]] does not extend to $n < 0$.
{{begin-eqn}}
{{eqn | l = 107\,928\,278\,317 + 0 \times 9\,922\,78... | Smallest 18 Primes in Arithmetic Sequence | https://proofwiki.org/wiki/Smallest_18_Primes_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Smallest_18_Primes_in_Arithmetic_Sequence | [
"Prime Numbers",
"Arithmetic Sequences"
] | [
"Definition:Prime Number",
"Definition:Arithmetic Sequence"
] | [
"Definition:Arithmetic Sequence",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime N... |
proofwiki-15291 | Prime Gap of 654 | There exists a prime gap of $654$ between $11\,000\,001\,446\,613\,353$ and $11\,000\,001\,446\,614\,007$. | $11\,000\,001\,446\,613\,353$ is a prime number.
$11\,000\,001\,446\,614\,007$ is a prime number.
It can be checked that all numbers between these two are composite.
{{qed}} | There exists a [[Definition:Prime Gap|prime gap]] of $654$ between $11\,000\,001\,446\,613\,353$ and $11\,000\,001\,446\,614\,007$. | $11\,000\,001\,446\,613\,353$ is a [[Definition:Prime Number|prime number]].
$11\,000\,001\,446\,614\,007$ is a [[Definition:Prime Number|prime number]].
It can be checked that all numbers between these two are [[Definition:Composite Number|composite]].
{{qed}} | Prime Gap of 654 | https://proofwiki.org/wiki/Prime_Gap_of_654 | https://proofwiki.org/wiki/Prime_Gap_of_654 | [
"Prime Gaps"
] | [
"Definition:Prime Gap"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Composite Number"
] |
proofwiki-15292 | Pair of Large Twin Primes | The integers defined as:
:$1\,159\,142\,985 \times 2^{2304} \pm 1$
are a pair of twin primes each with $703$ digits. | $1\,159\,142\,985 \times 2^{2304} - 1$:
{{Alpertron-factorizer|date = $6$th March $2022$|time = $0.2$ seconds}}
$1\,159\,142\,985 \times 2^{2304} + 1$:
{{Alpertron-factorizer|date = $6$th March $2022$|time = $0.8$ seconds}}
{{qed}} | The [[Definition:Integer|integers]] defined as:
:$1\,159\,142\,985 \times 2^{2304} \pm 1$
are a pair of [[Definition:Twin Primes|twin primes]] each with $703$ [[Definition:Digit|digits]]. | $1\,159\,142\,985 \times 2^{2304} - 1$:
{{Alpertron-factorizer|date = $6$th March $2022$|time = $0.2$ seconds}}
$1\,159\,142\,985 \times 2^{2304} + 1$:
{{Alpertron-factorizer|date = $6$th March $2022$|time = $0.8$ seconds}}
{{qed}} | Pair of Large Twin Primes | https://proofwiki.org/wiki/Pair_of_Large_Twin_Primes | https://proofwiki.org/wiki/Pair_of_Large_Twin_Primes | [
"Twin Primes/Examples"
] | [
"Definition:Integer",
"Definition:Twin Primes",
"Definition:Digit"
] | [] |
proofwiki-15293 | Integers under Subtraction do not form Group | Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction.
Then $\struct {\Z, -}$ is not a group. | It is to be demonstrated that $\struct {\Z, -}$ does not satisfy the group axioms.
First it is noted that Integer Subtraction is Closed.
Thus $\struct {\Z, -}$ fulfils {{Group-axiom|0}}.
However, we then have Subtraction on Numbers is Not Associative.
So, for example:
:$3 - \paren {2 - 1} = 2 \ne \paren {3 - 2} - 1 = 0... | Let $\struct {\Z, -}$ denote the [[Definition:Algebraic Structure with One Operation|algebraic structure]] formed by the set of [[Definition:Integer|integers]] under the [[Definition:Binary Operation|operation]] of [[Definition:Integer Subtraction|subtraction]].
Then $\struct {\Z, -}$ is not a [[Definition:Group|grou... | It is to be demonstrated that $\struct {\Z, -}$ does not satisfy the [[Axiom:Group Axioms|group axioms]].
First it is noted that [[Integer Subtraction is Closed]].
Thus $\struct {\Z, -}$ fulfils {{Group-axiom|0}}.
However, we then have [[Subtraction on Numbers is Not Associative]].
So, for example:
:$3 - \paren {2... | Integers under Subtraction do not form Group | https://proofwiki.org/wiki/Integers_under_Subtraction_do_not_form_Group | https://proofwiki.org/wiki/Integers_under_Subtraction_do_not_form_Group | [
"Integer Subtraction",
"Examples of Groups"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Integer",
"Definition:Operation/Binary Operation",
"Definition:Subtraction/Integers",
"Definition:Group"
] | [
"Axiom:Group Axioms",
"Integer Subtraction is Closed",
"Subtraction on Numbers is Not Associative",
"Axiom:Group Axioms"
] |
proofwiki-15294 | Sequence of Powers of Number less than One/Necessary Condition | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a null sequence. | {{WLOG}}, assume that $x \ne 0$.
Observe that {{hypothesis}}:
:$0 < \size x < 1$
Thus by Ordering of Reciprocals:
:$\size x^{-1} > 1$
Define:
:$h = \size x^{-1} - 1 > 0$
Then:
:$x = \dfrac 1 {1 + h}$
By the binomial theorem, we have that:
:$\paren {1 + h}^n = 1 + n h + \cdots + h^n > n h$
because $h > 0$.
By Absolute V... | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a [[Definition:Real Null Sequence|null sequence]]. | {{WLOG}}, assume that $x \ne 0$.
Observe that {{hypothesis}}:
:$0 < \size x < 1$
Thus by [[Ordering of Reciprocals]]:
:$\size x^{-1} > 1$
Define:
:$h = \size x^{-1} - 1 > 0$
Then:
:$x = \dfrac 1 {1 + h}$
By the [[Binomial Theorem|binomial theorem]], we have that:
:$\paren {1 + h}^n = 1 + n h + \cdots + h^n > n h... | Sequence of Powers of Number less than One/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition/Proof_1 | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Real Sequence",
"Definition:Null Sequence/Real Numbers"
] | [
"Ordering of Reciprocals",
"Binomial Theorem",
"Absolute Value Function is Completely Multiplicative",
"Sequence of Powers of Reciprocals is Null Sequence/Corollary",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Definition:Limit of Sequence (Number Field)"
] |
proofwiki-15295 | Sequence of Powers of Number less than One/Necessary Condition | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a null sequence. | Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Suppose that:
:$\exists N \in \N: \size x^N < \epsilon$
Then the result follows by the definition of a limit, because:
:$\forall n \in \N: n \ge N \implies \size {x^n} = \size x^n \le \size x^N < \epsilon$
where Absolute Value Function is Completely Multipl... | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a [[Definition:Real Null Sequence|null sequence]]. | Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Suppose that:
:$\exists N \in \N: \size x^N < \epsilon$
Then the result follows by the definition of a [[Definition:Limit of Sequence (Number Field)|limit]], because:
:$\forall n \in \N: n \ge N \implies \size... | Sequence of Powers of Number less than One/Necessary Condition/Proof 2 | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition/Proof_2 | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Real Sequence",
"Definition:Null Sequence/Real Numbers"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Limit of Sequence (Number Field)",
"Absolute Value Function is Completely Multiplicative",
"Axiom of Archimedes",
"Definition:Natural Numbers",
"Sum of Geometric Sequence",
"Definition:Contradiction"
] |
proofwiki-15296 | Sequence of Powers of Number less than One/Necessary Condition | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a null sequence. | Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
By the Axiom of Archimedes, there exists a natural number $M$ such that:
:$M > \dfrac 1 {\paren {1 - \size x} \epsilon}$
By the Well-Ordering Principle, there exists a smallest natural number $m$ such that:
:$\exists N \in \N: m > M \size x^N$
Note that:
:$... | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a [[Definition:Real Null Sequence|null sequence]]. | Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
By the [[Axiom of Archimedes]], there exists a [[Definition:Natural Numbers|natural number]] $M$ such that:
:$M > \dfrac 1 {\paren {1 - \size x} \epsilon}$
By the [[Well-Ordering Principle]], there exists a [[... | Sequence of Powers of Number less than One/Necessary Condition/Proof 3 | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition/Proof_3 | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Real Sequence",
"Definition:Null Sequence/Real Numbers"
] | [
"Definition:Strictly Positive/Real Number",
"Axiom of Archimedes",
"Definition:Natural Numbers",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Natural Numbers",
"Absolute Value Function is Completely Multiplicative",
"Definition:Limit of Sequence (Number Field)"
] |
proofwiki-15297 | Sequence of Powers of Number less than One/Necessary Condition | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a null sequence. | Define:
:$\ds L = \inf_{n \mathop \in \N} \size x^n$
By the Continuum Property, such an $L$ exists in $\R$.
Clearly, $L \ge 0$.
{{AimForCont}} $L > 0$.
Then, by the definition of the infimum, we can choose $n \in \N$ such that $\size x^n < L \size x^{-1}$.
But then $\size x^{n + 1} < L$, which contradicts the definitio... | Let $x \in \R$ be such that $\size{x} < 1$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then $\sequence {x_n}$ is a [[Definition:Real Null Sequence|null sequence]]. | Define:
:$\ds L = \inf_{n \mathop \in \N} \size x^n$
By the [[Continuum Property]], such an $L$ exists in $\R$.
Clearly, $L \ge 0$.
{{AimForCont}} $L > 0$.
Then, by the definition of the [[Definition:Infimum of Set|infimum]], we can choose $n \in \N$ such that $\size x^n < L \size x^{-1}$.
But then $\size x^{n + ... | Sequence of Powers of Number less than One/Necessary Condition/Proof 4 | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Necessary_Condition/Proof_4 | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Real Sequence",
"Definition:Null Sequence/Real Numbers"
] | [
"Continuum Property",
"Definition:Infimum of Set",
"Definition:Contradiction",
"Definition:Strictly Positive/Real Number",
"Definition:Infimum of Set",
"Absolute Value Function is Completely Multiplicative",
"Definition:Limit of Sequence (Number Field)"
] |
proofwiki-15298 | Odd Integers under Multiplication do not form Group | Let $S$ be the set of odd integers:
:$S = \set {x \in \Z: \exists n \in \Z: x = 2 n + 1}$
Let $\struct {S, \times}$ denote the algebraic structure formed by $S$ under the operation of multiplication.
Then $\struct {S, \times}$ is not a group. | It is to be demonstrated that $\struct {S, \times}$ does not satisfy the group axioms.
First it is noted that Integer Multiplication is Closed.
Then from Odd Number multiplied by Odd Number is Odd, $S$ is closed under $\times$.
Thus $\struct {S, \times}$ fulfils {{Group-axiom|0}}.
From Integer Multiplication is Associa... | Let $S$ be the [[Definition:Set|set]] of [[Definition:Odd Integer|odd integers]]:
:$S = \set {x \in \Z: \exists n \in \Z: x = 2 n + 1}$
Let $\struct {S, \times}$ denote the [[Definition:Algebraic Structure with One Operation|algebraic structure]] formed by $S$ under the [[Definition:Binary Operation|operation]] of [[D... | It is to be demonstrated that $\struct {S, \times}$ does not satisfy the [[Axiom:Group Axioms|group axioms]].
First it is noted that [[Integer Multiplication is Closed]].
Then from [[Odd Number multiplied by Odd Number is Odd]], $S$ is [[Definition:Closed Algebraic Structure|closed]] under $\times$.
Thus $\struct {... | Odd Integers under Multiplication do not form Group | https://proofwiki.org/wiki/Odd_Integers_under_Multiplication_do_not_form_Group | https://proofwiki.org/wiki/Odd_Integers_under_Multiplication_do_not_form_Group | [
"Odd Integers",
"Integer Multiplication",
"Examples of Groups"
] | [
"Definition:Set",
"Definition:Odd Integer",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Multiplication/Integers",
"Definition:Group"
] | [
"Axiom:Group Axioms",
"Integer Multiplication is Closed",
"Odd Number multiplied by Odd Number is Odd",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Integer Multiplication is Associative",
"Definition:Associative Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
... |
proofwiki-15299 | Sequence of Powers of Number less than One/Sufficient Condition | Let $x \in \R$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Let $\sequence {x_n}$ be a null sequence.
Then $\size x < 1$. | By Reciprocal of Null Sequence:
:$\sequence {x_n}$ converges to $0$ {{iff}} $\sequence {\dfrac 1 {x_n} }$ diverges to $\infty$.
By the definition of divergence to $\infty$:
:$\exists N \in \N: \forall n \ge N: \size {\dfrac 1 {x_n} } > 1$
In particular:
:$\size {\dfrac 1 {x_N} } > 1$
By Ordering of Reciprocals:
:$\size... | Let $x \in \R$.
Let $\sequence {x_n}$ be the [[Definition:Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Let $\sequence {x_n}$ be a [[Definition:Real Null Sequence|null sequence]].
Then $\size x < 1$. | By [[Reciprocal of Null Sequence]]:
:$\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]] to $0$ {{iff}} $\sequence {\dfrac 1 {x_n} }$ [[Definition:Unbounded Divergent Real Sequence|diverges to $\infty$]].
By the definition of [[Definition:Unbounded Divergent Real Sequence|divergence to $\infty$]]:
:$\e... | Sequence of Powers of Number less than One/Sufficient Condition | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Sufficient_Condition | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Sufficient_Condition | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Sequence",
"Definition:Null Sequence/Real Numbers"
] | [
"Reciprocal of Null Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Unbounded Divergent Sequence/Real Sequence",
"Definition:Unbounded Divergent Sequence/Real Sequence",
"Ordering of Reciprocals",
"Inequality of Product of Unequal Numbers",
"Definition:Contradiction"
] |
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