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-15300 | Square Matrices with +1 or -1 Determinant under Multiplication forms Group | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $S$ be the set of square matrices of order $n$ of real numbers whose determinant is either $1$ or $-1$.
Let $\struct {S, \times}$ denote the algebraic structure formed by $S$ whose operation is (conventional) matrix multiplication.
Then $\struct {S, \times}$ is a ... | Taking the group axioms in turn: | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $S$ be the [[Definition:Set|set]] of [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order]] $n$ of [[Definition:Real Numbers|real numbers]] whose [[Definition:Determinant of Matrix|dete... | Taking the [[Axiom:Group Axioms|group axioms]] in turn: | Square Matrices with +1 or -1 Determinant under Multiplication forms Group | https://proofwiki.org/wiki/Square_Matrices_with_+1_or_-1_Determinant_under_Multiplication_forms_Group | https://proofwiki.org/wiki/Square_Matrices_with_+1_or_-1_Determinant_under_Multiplication_forms_Group | [
"Examples of Groups",
"Matrix Groups"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Real Number",
"Definition:Determinant/Matrix",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Matrix... | [
"Axiom:Group Axioms",
"Axiom:Group Axioms"
] |
proofwiki-15301 | Real Sine Function is neither Injective nor Surjective | The real sine function is neither an injection nor a surjection. | This is immediately apparent from the graph of the sine function:
{{:Graph of Sine Function}}
For example:
:$\map \sin 0 = \map \sin \pi = 0$
and so the real sine function is not an injection.
Then, for example:
:$\nexists x \in \R: \map \sin x = 2$
and so the real sine function is not a surjection.
{{qed}} | The [[Definition:Real Sine Function|real sine function]] is neither an [[Definition:Injection|injection]] nor a [[Definition:Surjection|surjection]]. | This is immediately apparent from the [[Graph of Sine Function|graph of the sine function]]:
{{:Graph of Sine Function}}
For example:
:$\map \sin 0 = \map \sin \pi = 0$
and so the [[Definition:Real Sine Function|real sine function]] is not an [[Definition:Injection|injection]].
Then, for example:
:$\nexists x \in \R... | Real Sine Function is neither Injective nor Surjective | https://proofwiki.org/wiki/Real_Sine_Function_is_neither_Injective_nor_Surjective | https://proofwiki.org/wiki/Real_Sine_Function_is_neither_Injective_nor_Surjective | [
"Sine Function",
"Examples of Injections",
"Examples of Surjections"
] | [
"Definition:Sine/Real Function",
"Definition:Injection",
"Definition:Surjection"
] | [
"Shape of Sine Function/Graph",
"Definition:Sine/Real Function",
"Definition:Injection",
"Definition:Sine/Real Function",
"Definition:Surjection"
] |
proofwiki-15302 | Sequence of Powers of Number less than One/Complex Numbers | Let $z \in \C$.
Let $\sequence {z_n}$ be the sequence in $\C$ defined as $z_n = z^n$.
Then:
:$\size z < 1$ {{iff}} $\sequence {z_n}$ is a null sequence. | By the definition of convergence:
:$\ds \lim_{n \mathop \to \infty} z_n = 0 \iff \lim_{n \mathop \to \infty} \size {z_n} = 0$
By Modulus of Product:
:$\forall n \in \N: \size {z_n} = \size {z^n} = \size z^n$
So:
:$\ds \lim_{n \mathop \to \infty} \size {z_n} = 0 \iff \lim_{n \mathop \to \infty} \size z^n = 0$
Since $\si... | Let $z \in \C$.
Let $\sequence {z_n}$ be the [[Definition:Complex Sequence|sequence in $\C$]] defined as $z_n = z^n$.
Then:
:$\size z < 1$ {{iff}} $\sequence {z_n}$ is a [[Definition:Complex Null Sequence|null sequence]]. | By the definition of [[Definition:Convergent Complex Sequence|convergence]]:
:$\ds \lim_{n \mathop \to \infty} z_n = 0 \iff \lim_{n \mathop \to \infty} \size {z_n} = 0$
By [[Modulus of Product]]:
:$\forall n \in \N: \size {z_n} = \size {z^n} = \size z^n$
So:
:$\ds \lim_{n \mathop \to \infty} \size {z_n} = 0 \iff \lim... | Sequence of Powers of Number less than One/Complex Numbers | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Complex_Numbers | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Complex_Numbers | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Complex Sequence",
"Definition:Null Sequence/Complex Numbers"
] | [
"Definition:Convergent Sequence/Complex Numbers",
"Complex Modulus of Product of Complex Numbers",
"Sequence of Powers of Number less than One"
] |
proofwiki-15303 | Sequence of Powers of Number less than One/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring
Let $x \in R$.
Let $\sequence {x_n}$ be the sequence in $R$ defined as $x_n = x^n$.
Then:
:$\norm x < 1$ {{iff}} $\sequence {x_n}$ is a null sequence. | Let $0_R$ be the zero of $R$.
By the definition of convergence:
:$\ds \lim_{n \mathop \to \infty} x_n = 0_R \iff \lim_{n \mathop \to \infty} \norm {x_n} = 0$
By {{Norm-axiom-mult|2}} then for each $n \in \N$:
:$\norm {x_n} = \norm {x^n} = \norm x^n$.
So:
:$\ds \lim_{n \mathop \to \infty} \norm {x_n} = 0 \iff \lim_{n \m... | Let $\struct {R, \norm {\,\cdot\,}}$ be a [[Definition:Normed Division Ring|normed division ring]]
Let $x \in R$.
Let $\sequence {x_n}$ be the [[Definition:Sequence|sequence]] in $R$ defined as $x_n = x^n$.
Then:
:$\norm x < 1$ {{iff}} $\sequence {x_n}$ is a [[Definition:Null Sequence in Normed Division Ring|null s... | Let $0_R$ be the [[Definition:Ring Zero|zero]] of $R$.
By the definition of [[Definition:Convergent Sequence in Normed Division Ring|convergence]]:
:$\ds \lim_{n \mathop \to \infty} x_n = 0_R \iff \lim_{n \mathop \to \infty} \norm {x_n} = 0$
By {{Norm-axiom-mult|2}} then for each $n \in \N$:
:$\norm {x_n} = \norm {x^n... | Sequence of Powers of Number less than One/Normed Division Ring | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Normed_Division_Ring | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Normed_Division_Ring | [
"Limits of Sequences",
"Sequence of Powers of Number less than One",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Null Sequence/Normed Division Ring"
] | [
"Definition:Ring Zero",
"Definition:Convergent Sequence/Normed Division Ring",
"Sequence of Powers of Number less than One"
] |
proofwiki-15304 | Sequence of Powers of Number less than One/Rational Numbers | Let $x \in \Q$.
Let $\sequence {x_n}$ be the sequence in $\Q$ defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ is a null sequence. | By the definition of convergence of a rational sequence:
:$\sequence {x_n}$ is a null sequence in the rational numbers {{iff}} $\sequence {x_n}$ is a null sequence in the real numbers
By Sequence of Powers of Real Number less than One:
:$\sequence {x_n}$ is a null sequence in the real numbers {{iff}} $\size x < 1$
{{qe... | Let $x \in \Q$.
Let $\sequence {x_n}$ be the [[Definition:Rational Sequence|sequence in $\Q$]] defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ is a [[Definition:Rational Null Sequence|null sequence]]. | By the definition of [[Definition:Convergent Rational Sequence|convergence of a rational sequence]]:
:$\sequence {x_n}$ is a [[Definition:Rational Null Sequence|null sequence in the rational numbers]] {{iff}} $\sequence {x_n}$ is a [[Definition:Real Null Sequence|null sequence in the real numbers]]
By [[Sequence of Po... | Sequence of Powers of Number less than One/Rational Numbers | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Rational_Numbers | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One/Rational_Numbers | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Rational Sequence",
"Definition:Null Sequence/Rational Numbers"
] | [
"Definition:Convergent Sequence/Rational Numbers",
"Definition:Null Sequence/Rational Numbers",
"Definition:Null Sequence/Real Numbers",
"Sequence of Powers of Number less than One",
"Definition:Null Sequence/Real Numbers",
"Category:Limits of Sequences",
"Category:Sequence of Powers of Number less than... |
proofwiki-15305 | Composite of Injection on Surjection is not necessarily Either | Let $f$ be an injection.
Let $g$ be a surjection.
Let $f \circ g$ denote the composition of $f$ with $g$.
Then it is not necessarily the case that $f \circ g$ is either a surjection or an injection. | Let $X, Y, Z$ be sets defined as:
{{begin-eqn}}
{{eqn | l = X
| r = \set {a, b, c}
| c =
}}
{{eqn | l = Y
| r = \set {1, 2}
| c =
}}
{{eqn | l = Z
| r = \set {z, y, z}
| c =
}}
{{end-eqn}}
Let $g: X \to Y$ be defined in two-row notation as:
:$\dbinom {a \ b \ c } {1 \ 2 \ 2}$
whic... | Let $f$ be an [[Definition:Injection|injection]].
Let $g$ be a [[Definition:Surjection|surjection]].
Let $f \circ g$ denote the [[Definition:Composition of Mappings|composition]] of $f$ with $g$.
Then it is not necessarily the case that $f \circ g$ is either a [[Definition:Surjection|surjection]] or an [[Definition... | Let $X, Y, Z$ be [[Definition:Set|sets]] defined as:
{{begin-eqn}}
{{eqn | l = X
| r = \set {a, b, c}
| c =
}}
{{eqn | l = Y
| r = \set {1, 2}
| c =
}}
{{eqn | l = Z
| r = \set {z, y, z}
| c =
}}
{{end-eqn}}
Let $g: X \to Y$ be defined in [[Definition:Two-Row Notation|two-row ... | Composite of Injection on Surjection is not necessarily Either | https://proofwiki.org/wiki/Composite_of_Injection_on_Surjection_is_not_necessarily_Either | https://proofwiki.org/wiki/Composite_of_Injection_on_Surjection_is_not_necessarily_Either | [
"Injections",
"Surjections",
"Composite Mappings"
] | [
"Definition:Injection",
"Definition:Surjection",
"Definition:Composition of Mappings",
"Definition:Surjection",
"Definition:Injection"
] | [
"Definition:Set",
"Definition:Permutation on n Letters/Two-Row Notation",
"Definition:Surjection",
"Definition:Permutation on n Letters/Two-Row Notation",
"Definition:Injection",
"Definition:Composition of Mappings",
"Definition:Injection",
"Definition:Surjection"
] |
proofwiki-15306 | Group of Order 3 is Unique | There exists exactly $1$ group of order $3$, up to isomorphism:
:$C_3$, the cyclic group of order $3$. | From Existence of Cyclic Group of Order n we have that one such group of order $3$ is the cyclic group of order $3$.
This is exemplified by the additive group of integers modulo $3$, whose Cayley table can be presented as:
{{:Modulo Addition/Cayley Table/Modulo 3}}{{qed|lemma}}
Consider an arbitrary group $\struct {G, ... | There exists exactly $1$ [[Definition:Group|group]] of [[Definition:Order of Group|order]] $3$, up to [[Definition:Group Isomorphism|isomorphism]]:
:$C_3$, the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $3$. | From [[Existence of Cyclic Group of Order n]] we have that one such [[Definition:Group|group]] of [[Definition:Order of Group|order]] $3$ is the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $3$.
This is exemplified by the [[Definition:Additive Group of Integers Modulo m|additive grou... | Group of Order 3 is Unique | https://proofwiki.org/wiki/Group_of_Order_3_is_Unique | https://proofwiki.org/wiki/Group_of_Order_3_is_Unique | [
"Groups of Order 3"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Cyclic Group",
"Definition:Order of Structure"
] | [
"Existence of Cyclic Group of Order n",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Additive Group of Integers Modulo m",
"Modulo Addition/Cayley Table/Modulo 3",
"Definition:Group",
"Definition:Identity (Abstract Algeb... |
proofwiki-15307 | Sequence of Powers of Number less than One | Let $x \in \R$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ is a null sequence. | === Necessary Condition ===
{{refactor|level = basic}}
[For other proofs of the Necessary Condition visit here.]
{{:Sequence of Powers of Number less than One/Necessary Condition/Proof 1}}{{qed|lemma}} | Let $x \in \R$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ is a [[Definition:Real Null Sequence|null sequence]]. | === [[Sequence of Powers of Number less than One/Necessary Condition|Necessary Condition]] ===
{{refactor|level = basic}}
[For other proofs of the [[Sequence of Powers of Number less than One/Necessary Condition|Necessary Condition]] visit [[Sequence of Powers of Number less than One/Necessary Condition|here]].]
{{:... | Sequence of Powers of Number less than One | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One | https://proofwiki.org/wiki/Sequence_of_Powers_of_Number_less_than_One | [
"Limits of Sequences",
"Sequence of Powers of Number less than One"
] | [
"Definition:Real Sequence",
"Definition:Null Sequence/Real Numbers"
] | [
"Sequence of Powers of Number less than One/Necessary Condition",
"Sequence of Powers of Number less than One/Necessary Condition",
"Sequence of Powers of Number less than One/Necessary Condition"
] |
proofwiki-15308 | Sequence of Powers of Number less than One | Let $x \in \R$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 $x \in \R$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 | 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-15309 | Sequence of Powers of Number less than One | Let $x \in \R$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 $x \in \R$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 | 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-15310 | Sequence of Powers of Number less than One | Let $x \in \R$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 $x \in \R$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 | 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-15311 | Sequence of Powers of Number less than One | Let $x \in \R$.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 $x \in \R$.
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as $x_n = x^n$.
Then:
:$\size x < 1$ {{iff}} $\sequence {x_n}$ 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 | 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-15312 | Set of Rotations is Subgroup of Symmetry Group | Let $G$ be a symmetry group.
Let $H$ be the subset of $G$ consisting of the rotations in $G$ about a given axis.
Then $H$ is a subgroup of $G$. | {{ProofWanted|Needs a more formal definition of rotation. Surprised this hasn't already been covered properly.}} | Let $G$ be a [[Definition:Symmetry Group|symmetry group]].
Let $H$ be the [[Definition:Subset|subset]] of $G$ consisting of the [[Definition:Rotation (Geometry)|rotations]] in $G$ about a given [[Definition:Axis of Rotation|axis]].
Then $H$ is a [[Definition:Subgroup|subgroup]] of $G$. | {{ProofWanted|Needs a more formal definition of rotation. Surprised this hasn't already been covered properly.}} | Set of Rotations is Subgroup of Symmetry Group | https://proofwiki.org/wiki/Set_of_Rotations_is_Subgroup_of_Symmetry_Group | https://proofwiki.org/wiki/Set_of_Rotations_is_Subgroup_of_Symmetry_Group | [
"Symmetry Groups"
] | [
"Definition:Symmetry Group",
"Definition:Subset",
"Definition:Rotation (Geometry)",
"Definition:Rotation (Geometry)/Axis",
"Definition:Subgroup"
] | [] |
proofwiki-15313 | Equivalence of Definitions of Equivalent Division Ring Norms | Let $R$ be a division ring.
Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be norms on $R$.
Let $d_1$ and $d_2$ be the metrics induced by the norms $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively.
{{TFAE|def = Equivalent Division Ring Norms}} | === Topologically Equivalent implies Convergently Equivalent ===
{{:Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent}}{{qed|lemma}} | Let $R$ be a [[Definition:Division Ring|division ring]].
Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be [[Definition:Norm on Division Ring|norms]] on $R$.
Let $d_1$ and $d_2$ be the [[Definition:Metric Induced by Norm|metrics induced]] by the [[Definition:Norm on Divisi... | === [[Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent|Topologically Equivalent implies Convergently Equivalent]] ===
{{:Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent}}{{qed|lemma... | Equivalence of Definitions of Equivalent Division Ring Norms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms | [
"Normed Division Rings",
"Norm Theory",
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Division Ring",
"Definition:Norm/Division Ring",
"Definition:Metric Induced by Norm",
"Definition:Norm/Division Ring"
] | [
"Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent"
] |
proofwiki-15314 | Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent | Let $d_1$ and $d_2$ be topologically equivalent metrics.
Then:
:$d_1$ and $d_2$ are convergently equivalent metrics. | Let $\sequence {x_n}$ converge to $l$ in $\norm {\, \cdot \,}_1$.
Let $\epsilon \in \R_{> 0}$ be given.
Let $\map {B_\epsilon^2} i$ denote the open ball centered on $l$ of radius $\epsilon$ in $\struct {R, \norm {\, \cdot \,}_2}$.
By Open Ball of Metric Space is Open Set then $\map {B_\epsilon^2} l$ is open set in $\st... | Let $d_1$ and $d_2$ be [[Definition:Topologically Equivalent Metrics|topologically equivalent metrics]].
Then:
:$d_1$ and $d_2$ are [[Definition:Equivalent Metrics|convergently equivalent metrics]]. | Let $\sequence {x_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converge]] to $l$ in $\norm {\, \cdot \,}_1$.
Let $\epsilon \in \R_{> 0}$ be given.
Let $\map {B_\epsilon^2} i$ denote the [[Definition:Open Ball|open ball]] [[Definition:Center of Open Ball|centered]] on $l$ of [[Definition:Radius of Ope... | Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Topologically_Equivalent_implies_Convergently_Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Topologically_Equivalent_implies_Convergently_Equivalent | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Topologically Equivalent Metrics",
"Definition:Equivalent Metrics"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Open Ball is Open Set/Pseudometric Space",
"Definition:Open Set/Metric Space",
"Definition:Topologically Equivalent Metrics",
"Definition:Open Set/Metric Space"... |
proofwiki-15315 | Equivalence of Definitions of Equivalent Division Ring Norms/Convergently Equivalent implies Null Sequence Equivalent | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ converges to $l$ in $\norm{\, \cdot \,}_1 \iff \sequence {x_n}$ is a converges to $l$ in $\norm {\, \cdot \,}_2$
Then for all sequences $\sequence {x_n}$ in $R$:
:$\sequence {x_n}$ is a null sequ... | Let $0_R$ be the zero of $R$, then:
:$\sequence {x_n}$ converges to $0_R$ in $\norm {\, \cdot \,}_1 \iff \sequence {x_n}$ converges to $0_R$ in $\norm {\, \cdot \,}_2$
Hence:
:$\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_2$
{{qed}} | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converges]] to $l$ in $\norm{\, \cdot \,}_1 \iff \sequence {x_n}$ is a [[Definition:Convergent Sequence in Normed Division Ring|converges]... | Let $0_R$ be the [[Definition:Ring Zero|zero]] of $R$, then:
:$\sequence {x_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converges]] to $0_R$ in $\norm {\, \cdot \,}_1 \iff \sequence {x_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converges]] to $0_R$ in $\norm {\, \cdot \,}_2$
Hence:
:... | Equivalence of Definitions of Equivalent Division Ring Norms/Convergently Equivalent implies Null Sequence Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Convergently_Equivalent_implies_Null_Sequence_Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Convergently_Equivalent_implies_Null_Sequence_Equivalent | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring"
] | [
"Definition:Ring Zero",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring"
] |
proofwiki-15316 | Equivalence of Definitions of Equivalent Division Ring Norms/Null Sequence Equivalent implies Open Unit Ball Equivalent | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_2$
Then $\forall x \in R$:
:$\norm x_1 < 1 \iff \norm x_2 < 1$ | Let $x \in R$.
Let $\sequence {x_n}$ be the sequence defined by: $\forall n: x_n = x^n$.
{{begin-eqn}}
{{eqn | l = \norm x_1 < 1 \quad
| o = \leadstoandfrom
| c = $\sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_1$
| cc= Sequence of Powers of Number less than One in Normed Division Ring
}}... | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ is a [[Definition:Null Sequence in Normed Division Ring|null sequence]] in $\norm {\, \cdot \,}_1 \iff \sequence {x_n}$ is a [[Definition:Null Sequence in Normed Division Ring|null sequence]] in... | Let $x \in R$.
Let $\sequence {x_n}$ be the [[Definition:Sequence|sequence]] defined by: $\forall n: x_n = x^n$.
{{begin-eqn}}
{{eqn | l = \norm x_1 < 1 \quad
| o = \leadstoandfrom
| c = $\sequence {x_n}$ is a [[Definition:Null Sequence in Normed Division Ring|null sequence]] in $\norm {\, \cdot \,}_1$
... | Equivalence of Definitions of Equivalent Division Ring Norms/Null Sequence Equivalent implies Open Unit Ball Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Null_Sequence_Equivalent_implies_Open_Unit_Ball_Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Null_Sequence_Equivalent_implies_Open_Unit_Ball_Equivalent | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Null Sequence/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring"
] | [
"Definition:Sequence",
"Definition:Null Sequence/Normed Division Ring",
"Sequence of Powers of Number less than One/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring",
"Sequence of Powers of Number less than One/Normed Division Ring"
] |
proofwiki-15317 | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:$\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$
Then:
:$\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$ | ==== Case 1 ====
For all $x \in R: x \ne 0_R$, let $x$ satisfy $\norm x_1 \ge 1$. | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:$\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$
Then:
:$\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$ | ==== Case 1 ====
For all $x \in R: x \ne 0_R$, let $x$ satisfy $\norm x_1 \ge 1$. | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [] | [] |
proofwiki-15318 | Product of Subgroups of Prime Power Order | Let $p$ be a prime number.
Let $G$ be a group of order $p^a k$, where:
:$a \in \Z_{>0}$ is a (strictly) positive integer
:$p$ is not a divisor of $k$.
Let $P \le G$ be a subgroup of $G$ of order $p^a$.
Let $Q \le G$ be a subgroup of $G$ of order $p^b$, where $0 < b \le a$.
Let it be the case that $Q$ is not a subgroup ... | From Intersection of Subgroups is Subgroup, $P \cap Q$ is a subgroup of $P$.
Thus:
:$\order {P \cap Q} = p^c$ for some $c \in \Z$ such that $0 \le c \le a$
where $\order {P \cap Q}$ denotes the order of $P \cap Q$.
We have:
{{begin-eqn}}
{{eqn | l = \order {P Q}
| r = \frac {\order P \order Q} {\order {P \cap Q} ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $p^a k$, where:
:$a \in \Z_{>0}$ is a [[Definition:Strictly Positive Integer|(strictly) positive integer]]
:$p$ is not a [[Definition:Divisor of Integer|divisor]] of $k$.
Let $P \le G$... | From [[Intersection of Subgroups is Subgroup]], $P \cap Q$ is a [[Definition:Subgroup|subgroup]] of $P$.
Thus:
:$\order {P \cap Q} = p^c$ for some $c \in \Z$ such that $0 \le c \le a$
where $\order {P \cap Q}$ denotes the [[Definition:Order of Group|order]] of $P \cap Q$.
We have:
{{begin-eqn}}
{{eqn | l = \order {P... | Product of Subgroups of Prime Power Order | https://proofwiki.org/wiki/Product_of_Subgroups_of_Prime_Power_Order | https://proofwiki.org/wiki/Product_of_Subgroups_of_Prime_Power_Order | [
"Subgroups"
] | [
"Definition:Prime Number",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Strictly Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Subgroup",
"... | [
"Intersection of Subgroups is Subgroup",
"Definition:Subgroup",
"Definition:Order of Structure",
"Order of Subgroup Product",
"Definition:Subgroup",
"Lagrange's Theorem (Group Theory)",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer"... |
proofwiki-15319 | Basis Theorem | $H$ is a basis for $E$ {{iff}} it contains exactly $n$ elements. | By hypothesis, let $H$ be a linearly independent subset of $E$ | $H$ is a [[Definition:Basis of Vector Space|basis]] for $E$ {{iff}} it contains exactly $n$ [[Definition:Element|elements]]. | [[Definition:By Hypothesis|By hypothesis]], let $H$ be a [[Definition:Linearly Independent Set|linearly independent subset]] of $E$ | Basis Theorem | https://proofwiki.org/wiki/Basis_Theorem | https://proofwiki.org/wiki/Basis_Theorem | [
"Bases of Vector Spaces",
"Dimension of Vector Space",
"Named Theorems"
] | [
"Definition:Basis of Vector Space",
"Definition:Element"
] | [
"Definition:By Hypothesis",
"Definition:Linearly Independent/Set"
] |
proofwiki-15320 | Generator of Vector Space is Basis iff Cardinality equals Dimension | :$G$ is a basis for $E$ {{iff}} $\card G = n$. | === Necessary Condition ===
Let $G$ be a basis for $E$.
From Cardinality of Basis of Vector Space, $\card G = n$.
{{qed|lemma}} | :$G$ is a [[Definition:Basis of Vector Space|basis]] for $E$ {{iff}} $\card G = n$. | === Necessary Condition ===
Let $G$ be a [[Definition:Basis of Vector Space|basis]] for $E$.
From [[Cardinality of Basis of Vector Space]], $\card G = n$.
{{qed|lemma}} | Generator of Vector Space is Basis iff Cardinality equals Dimension | https://proofwiki.org/wiki/Generator_of_Vector_Space_is_Basis_iff_Cardinality_equals_Dimension | https://proofwiki.org/wiki/Generator_of_Vector_Space_is_Basis_iff_Cardinality_equals_Dimension | [
"Generators of Vector Spaces",
"Bases of Vector Spaces",
"Dimension of Vector Space"
] | [
"Definition:Basis of Vector Space"
] | [
"Definition:Basis of Vector Space",
"Cardinality of Basis of Vector Space",
"Definition:Basis of Vector Space"
] |
proofwiki-15321 | ISBN-10 is Error-Correcting Code/Transposition Error | If any two of the first $9$ digits are transposed, the check digit will be wrong. | {{ProofWanted|Ongoing}}
Category:ISBN-10 is Error-Correcting Code
mwbruqai92ntf6l904fwb5runnoin1b | If any two of the first $9$ [[Definition:Digit|digits]] are transposed, the [[Definition:Check Digit|check digit]] will be wrong. | {{ProofWanted|Ongoing}}
[[Category:ISBN-10 is Error-Correcting Code]]
mwbruqai92ntf6l904fwb5runnoin1b | ISBN-10 is Error-Correcting Code/Transposition Error | https://proofwiki.org/wiki/ISBN-10_is_Error-Correcting_Code/Transposition_Error | https://proofwiki.org/wiki/ISBN-10_is_Error-Correcting_Code/Transposition_Error | [
"ISBN-10 is Error-Correcting Code"
] | [
"Definition:Digit",
"Definition:Check Digit"
] | [
"Category:ISBN-10 is Error-Correcting Code"
] |
proofwiki-15322 | ISBN-10 is Error-Correcting Code/Transmission Error | If an error has been made in any one of the first $9$ digits, the check digit will be wrong. | Let $S$ denote an ISBN-$10$ whose $k$th digit is $d_k$.
Let $d_S$ denote the check digit of $S$.
Let $S'$ denote the ISBN-$10$ $S$ whose $n$th digit has been transmitted incorrectly, as $d'_n$.
Let $d'_S$ denote the check digit calculated on $S'$ according to the algorithm via which calculated $d_S$ on $S$.
It will be ... | If an error has been made in any one of the first $9$ [[Definition:Digit|digits]], the [[Definition:Check Digit|check digit]] will be wrong. | Let $S$ denote an [[Definition:ISBN-10|ISBN-$10$]] whose $k$th [[Definition:Digit|digit]] is $d_k$.
Let $d_S$ denote the [[Definition:Check Digit|check digit]] of $S$.
Let $S'$ denote the [[Definition:ISBN-10|ISBN-$10$]] $S$ whose $n$th [[Definition:Digit|digit]] has been transmitted incorrectly, as $d'_n$.
Let $d'_... | ISBN-10 is Error-Correcting Code/Transmission Error | https://proofwiki.org/wiki/ISBN-10_is_Error-Correcting_Code/Transmission_Error | https://proofwiki.org/wiki/ISBN-10_is_Error-Correcting_Code/Transmission_Error | [
"ISBN-10 is Error-Correcting Code"
] | [
"Definition:Digit",
"Definition:Check Digit"
] | [
"Definition:International Standard Book Number/ISBN-10",
"Definition:Digit",
"Definition:Check Digit",
"Definition:International Standard Book Number/ISBN-10",
"Definition:Digit",
"Definition:Check Digit",
"Definition:International Standard Book Number/ISBN-10",
"Definition:Symbol",
"Definition:Chec... |
proofwiki-15323 | Cardinality of Master Code | Let $\map V {n, p}$ be a master code of length $n$ modulo $p$.
Then there are $p^n$ elements of $\map V {n, p}$. | For each term of a sequence in $\map V {n, p}$ there are $p$ possible values.
There are $n$ such terms.
Hence there are $\underbrace {p \times p \times \cdots \times p}_{n \text { times} } = p^n$ different possible sequences in $\map V {n, p}$.
{{qed}} | Let $\map V {n, p}$ be a [[Definition:Master Code|master code]] of [[Definition:Length of Sequence|length]] $n$ modulo $p$.
Then there are $p^n$ [[Definition:Element|elements]] of $\map V {n, p}$. | For each [[Definition:Term of Sequence|term]] of a [[Definition:Finite Sequence|sequence]] in $\map V {n, p}$ there are $p$ possible values.
There are $n$ such [[Definition:Term of Sequence|terms]].
Hence there are $\underbrace {p \times p \times \cdots \times p}_{n \text { times} } = p^n$ different possible [[Defini... | Cardinality of Master Code | https://proofwiki.org/wiki/Cardinality_of_Master_Code | https://proofwiki.org/wiki/Cardinality_of_Master_Code | [
"Linear Codes"
] | [
"Definition:Linear Code/Master Code",
"Definition:Length of Sequence",
"Definition:Element"
] | [
"Definition:Term of Sequence",
"Definition:Finite Sequence",
"Definition:Term of Sequence",
"Definition:Finite Sequence"
] |
proofwiki-15324 | Master Code forms Vector Space | Let $\map V {n, p}$ be a master code of length $n$ modulo $p$.
Then $\map V {n, p}$ forms a vector space over $\Z_p$ of $n$ dimensions. | Recall the vector space axioms:
{{:Axiom:Vector Space Axioms}}
First, the set of sequences $\tuple {x_1, x_2, \ldots, x_n}$, for $x_1, x_2, \ldots, x_n \in \Z_p$, has to be shown to fulfil the abelian group axioms.
This follows from:
:Integers Modulo m under Addition form Cyclic Group
and:
:Cyclic Group is Abelian.
{{P... | Let $\map V {n, p}$ be a [[Definition:Master Code|master code]] of [[Definition:Length of Sequence|length]] $n$ modulo $p$.
Then $\map V {n, p}$ forms a [[Definition:Vector Space|vector space]] over $\Z_p$ of [[Definition:Dimension of Vector Space|$n$ dimensions]]. | Recall the [[Axiom:Vector Space Axioms|vector space axioms]]:
{{:Axiom:Vector Space Axioms}}
First, the [[Definition:Set|set]] of [[Definition:Finite Sequence|sequences]] $\tuple {x_1, x_2, \ldots, x_n}$, for $x_1, x_2, \ldots, x_n \in \Z_p$, has to be shown to fulfil the [[Axiom:Abelian Group Axioms|abelian group axi... | Master Code forms Vector Space | https://proofwiki.org/wiki/Master_Code_forms_Vector_Space | https://proofwiki.org/wiki/Master_Code_forms_Vector_Space | [
"Linear Codes"
] | [
"Definition:Linear Code/Master Code",
"Definition:Length of Sequence",
"Definition:Vector Space",
"Definition:Dimension of Vector Space"
] | [
"Axiom:Vector Space Axioms",
"Definition:Set",
"Definition:Finite Sequence",
"Axiom:Abelian Group Axioms",
"Integers Modulo m under Addition form Cyclic Group",
"Cyclic Group is Abelian"
] |
proofwiki-15325 | Conditions Satisfied by Linear Code | Let $p$ be a prime number.
Let $\Z_p$ be the set of residue classes modulo $p$.
Let $C := \tuple {n, k}$ be a linear code of a master code $\map V {n, p}$.
Then $C$ satisfies the following conditions:
:$(C \, 1): \quad \forall \mathbf x, \mathbf y \in C: \mathbf x + \paren {-\mathbf y} \in C$
:$(C \, 2): \quad \forall ... | From Master Code forms Vector Space, $\map V {n, p}$ is a vector space.
By definition, $\tuple {n, k}$ is a subspace of $\map V {n, p}$.
The result follows by the fact that a subspace is itself a vector space.
{{finish|I lose patience with the fine detail.}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\Z_p$ be the [[Definition:Set of Residue Classes|set of residue classes modulo $p$]].
Let $C := \tuple {n, k}$ be a [[Definition:Linear Code|linear code]] of a [[Definition:Master Code|master code]] $\map V {n, p}$.
Then $C$ satisfies the following condit... | From [[Master Code forms Vector Space]], $\map V {n, p}$ is a [[Definition:Vector Space|vector space]].
By definition, $\tuple {n, k}$ is a [[Definition:Vector Subspace|subspace]] of $\map V {n, p}$.
The result follows by the fact that a [[Definition:Vector Subspace|subspace]] is itself a [[Definition:Vector Space|ve... | Conditions Satisfied by Linear Code | https://proofwiki.org/wiki/Conditions_Satisfied_by_Linear_Code | https://proofwiki.org/wiki/Conditions_Satisfied_by_Linear_Code | [
"Linear Codes"
] | [
"Definition:Prime Number",
"Definition:Set of Residue Classes",
"Definition:Linear Code",
"Definition:Linear Code/Master Code",
"Definition:Addition of Codewords in Linear Code",
"Definition:Multiple of Codeword in Linear Code"
] | [
"Master Code forms Vector Space",
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Vector Subspace",
"Definition:Vector Space"
] |
proofwiki-15326 | Hamming Distance is Distance Function | Let $\map V {n, p}$ be a master code.
Let $d: V \times V \to \Z$ be the mapping defined as:
:$\forall u, v \in V: \map d {u, v} =$ the Hamming distance between $u$ and $v$
that is, the number of corresponding terms at which $u$ and $v$ are different.
Then $d$ defines a distance function in the sense of a metric space. | It is to be demonstrated that $d$ satisfies all the metric space axioms.
Let $u, v, w \in \map V {n, p}$ be arbitrary. | Let $\map V {n, p}$ be a [[Definition:Master Code|master code]].
Let $d: V \times V \to \Z$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall u, v \in V: \map d {u, v} =$ the [[Definition:Hamming Distance|Hamming distance]] between $u$ and $v$
that is, the number of corresponding [[Definition:Term of Sequen... | It is to be demonstrated that $d$ satisfies all the [[Axiom:Metric Space Axioms|metric space axioms]].
Let $u, v, w \in \map V {n, p}$ be arbitrary. | Hamming Distance is Distance Function | https://proofwiki.org/wiki/Hamming_Distance_is_Distance_Function | https://proofwiki.org/wiki/Hamming_Distance_is_Distance_Function | [
"Hamming Distance"
] | [
"Definition:Linear Code/Master Code",
"Definition:Mapping",
"Definition:Hamming Distance",
"Definition:Term of Sequence",
"Definition:Distance Function",
"Definition:Metric Space"
] | [
"Axiom:Metric Space Axioms",
"Axiom:Metric Space Axioms"
] |
proofwiki-15327 | Minimum Distance of Linear Code is Smallest Weight of Non-Zero Codeword | Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$.
Let $\map d C$ denote the minimum distance of $C$.
Then:
:$\map d C = \ds \min_{u \mathop \in C} \map w u$
where $\map w u$ denotes the weight of $u$. | Let $f := \ds \min_{u \mathop \in C} \map w u$.
Let $\mathbf 0$ denote the codeword in $\map V {n, p}$ consisting of all zeroes.
As $C$ is a subspace of $\map V {n, p}$, we have that $\mathbf 0 \in C$.
Let $w$ be a codeword with weight $f$.
Then:
:$\map d {w, \mathbf 0} = f$
so $f \ge \map d C$.
Let $u, v \in C$ such t... | Let $C$ be a [[Definition:Linear Code|linear $\tuple {n, k}$-code]] whose [[Definition:Master Code|master code]] is $\map V {n, p}$.
Let $\map d C$ denote the [[Definition:Minimum Distance of Linear Code|minimum distance]] of $C$.
Then:
:$\map d C = \ds \min_{u \mathop \in C} \map w u$
where $\map w u$ denotes the ... | Let $f := \ds \min_{u \mathop \in C} \map w u$.
Let $\mathbf 0$ denote the [[Definition:Codeword of Linear Code|codeword]] in $\map V {n, p}$ consisting of all [[Definition:Zero Digit|zeroes]].
As $C$ is a [[Definition:Vector Subspace|subspace]] of $\map V {n, p}$, we have that $\mathbf 0 \in C$.
Let $w$ be a [[Def... | Minimum Distance of Linear Code is Smallest Weight of Non-Zero Codeword | https://proofwiki.org/wiki/Minimum_Distance_of_Linear_Code_is_Smallest_Weight_of_Non-Zero_Codeword | https://proofwiki.org/wiki/Minimum_Distance_of_Linear_Code_is_Smallest_Weight_of_Non-Zero_Codeword | [
"Linear Codes"
] | [
"Definition:Linear Code",
"Definition:Linear Code/Master Code",
"Definition:Minimum Distance of Linear Code",
"Definition:Weight of Linear Codeword"
] | [
"Definition:Linear Code/Codeword",
"Definition:Zero Digit",
"Definition:Vector Subspace",
"Definition:Linear Code/Codeword",
"Definition:Weight of Linear Codeword",
"Definition:Linear Code",
"Definition:Difference between Linear Codewords",
"Definition:Weight of Linear Codeword"
] |
proofwiki-15328 | Error Detection Capability of Linear Code | Let $C$ be a linear code.
Let $C$ have a minimum distance $d$.
Then $C$ detects $d - 1$ or fewer transmission errors. | Let $C$ be a linear code whose master code is $V$.
Let $c \in C$ be a transmitted codeword.
Let $v$ be the received word from $c$.
By definition, $v$ is an element of $V$.
Let $v$ have a Hamming distance $f$ from $c$, where $f \le d - 1$.
Thus there have been $f$ transmission errors.
As $d$ is the minimum distance it i... | Let $C$ be a [[Definition:Linear Code|linear code]].
Let $C$ have a [[Definition:Minimum Distance of Linear Code|minimum distance]] $d$.
Then $C$ detects $d - 1$ or fewer [[Definition:Transmission Error|transmission errors]]. | Let $C$ be a [[Definition:Linear Code|linear code]] whose [[Definition:Master Code|master code]] is $V$.
Let $c \in C$ be a [[Definition:Transmitted Codeword|transmitted codeword]].
Let $v$ be the [[Definition:Received Word|received word]] from $c$.
By definition, $v$ is an [[Definition:Element|element]] of $V$.
L... | Error Detection Capability of Linear Code | https://proofwiki.org/wiki/Error_Detection_Capability_of_Linear_Code | https://proofwiki.org/wiki/Error_Detection_Capability_of_Linear_Code | [
"Linear Codes"
] | [
"Definition:Linear Code",
"Definition:Minimum Distance of Linear Code",
"Definition:Transmission Error"
] | [
"Definition:Linear Code",
"Definition:Linear Code/Master Code",
"Definition:Transmitted Codeword",
"Definition:Received Word",
"Definition:Element",
"Definition:Hamming Distance",
"Definition:Transmission Error",
"Definition:Minimum Distance of Linear Code",
"Definition:Linear Code/Codeword",
"Def... |
proofwiki-15329 | Error Correction Capability of Linear Code | Let $C$ be a linear code.
Let $C$ have a minimum distance $d$.
Then $C$ corrects $e$ transmission errors for all $e$ such that $2 e + 1 \le d$. | Let $C$ be a linear code whose master code is $V$.
Let $c \in C$ be a transmitted codeword.
Let $v$ be the received word from $c$.
By definition, $v$ is an element of $V$.
Let $v$ have a Hamming distance $e$ from $c$, where $2 e + 1 \le d$.
Thus there have been $e$ transmission errors.
{{AimForCont}} $c_1$ is a codewor... | Let $C$ be a [[Definition:Linear Code|linear code]].
Let $C$ have a [[Definition:Minimum Distance of Linear Code|minimum distance]] $d$.
Then $C$ corrects $e$ [[Definition:Transmission Error|transmission errors]] for all $e$ such that $2 e + 1 \le d$. | Let $C$ be a [[Definition:Linear Code|linear code]] whose [[Definition:Master Code|master code]] is $V$.
Let $c \in C$ be a [[Definition:Transmitted Codeword|transmitted codeword]].
Let $v$ be the [[Definition:Received Word|received word]] from $c$.
By definition, $v$ is an [[Definition:Element|element]] of $V$.
L... | Error Correction Capability of Linear Code | https://proofwiki.org/wiki/Error_Correction_Capability_of_Linear_Code | https://proofwiki.org/wiki/Error_Correction_Capability_of_Linear_Code | [
"Linear Codes"
] | [
"Definition:Linear Code",
"Definition:Minimum Distance of Linear Code",
"Definition:Transmission Error"
] | [
"Definition:Linear Code",
"Definition:Linear Code/Master Code",
"Definition:Transmitted Codeword",
"Definition:Received Word",
"Definition:Element",
"Definition:Hamming Distance",
"Definition:Transmission Error",
"Definition:Linear Code/Codeword",
"Definition:Distinct/Plural",
"Definition:Hamming ... |
proofwiki-15330 | Golay Ternary Code has Minimum Distance 5 | The Golay ternary code has a minimum distance of $5$. | Let $C$ denote the Golay ternary code.
By inspection of the standard generator matrix $G$ of $C$:
:$G := \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\
0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 2 \\
0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 &... | The [[Definition:Golay Ternary Code|Golay ternary code]] has a [[Definition:Minimum Distance of Linear Code|minimum distance]] of $5$. | Let $C$ denote the [[Definition:Golay Ternary Code|Golay ternary code]].
By inspection of the [[Definition:Standard Generator Matrix for Linear Code|standard generator matrix]] $G$ of $C$:
:$G := \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\
0 & 0 & 1 & 0 & 0... | Golay Ternary Code has Minimum Distance 5 | https://proofwiki.org/wiki/Golay_Ternary_Code_has_Minimum_Distance_5 | https://proofwiki.org/wiki/Golay_Ternary_Code_has_Minimum_Distance_5 | [
"Golay Ternary Code"
] | [
"Definition:Golay Ternary Code",
"Definition:Minimum Distance of Linear Code"
] | [
"Definition:Golay Ternary Code",
"Definition:Standard Generator Matrix for Linear Code",
"Definition:Weight of Linear Codeword",
"Definition:Linear Code/Codeword",
"Definition:Minimum Distance of Linear Code",
"Definition:Minimum Distance of Linear Code"
] |
proofwiki-15331 | Golay Ternary Code Corrects 2 Errors | The Golay ternary code corrects $2$ transmission errors. | We have that Golay Ternary Code has Minimum Distance 5.
The result follows from Error Correction Capability of Linear Code.
{{qed}} | The [[Definition:Golay Ternary Code|Golay ternary code]] corrects $2$ [[Definition:Transmission Error|transmission errors]]. | We have that [[Golay Ternary Code has Minimum Distance 5]].
The result follows from [[Error Correction Capability of Linear Code]].
{{qed}} | Golay Ternary Code Corrects 2 Errors | https://proofwiki.org/wiki/Golay_Ternary_Code_Corrects_2_Errors | https://proofwiki.org/wiki/Golay_Ternary_Code_Corrects_2_Errors | [
"Golay Ternary Code"
] | [
"Definition:Golay Ternary Code",
"Definition:Transmission Error"
] | [
"Golay Ternary Code has Minimum Distance 5",
"Error Correction Capability of Linear Code"
] |
proofwiki-15332 | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 1 | :$\norm {\, \cdot \,}_1$ is the trivial norm. | We prove the contrapositive.
Let $\norm {\, \cdot \,}_1$ be a nontrivial norm.
Then:
:$\exists y \in R: \norm y_1 \ne 0, \norm y_1 \ne 1$.
By Real Numbers form Totally Ordered Field either $\norm y_1 < 1$ or $\norm y_1 > 1$.
Suppose $\norm y_1 > 1$.
By Norm axiom $(\text N 1)$: Positive Definiteness:
:$y \ne 0_R$
By No... | :$\norm {\, \cdot \,}_1$ is the [[Definition:Trivial Norm on Division Ring|trivial norm]]. | We prove the [[Definition:Contrapositive Statement|contrapositive]].
Let $\norm {\, \cdot \,}_1$ be a [[Definition:Nontrivial Division Ring Norm|nontrivial norm]].
Then:
:$\exists y \in R: \norm y_1 \ne 0, \norm y_1 \ne 1$.
By [[Real Numbers form Totally Ordered Field]] either $\norm y_1 < 1$ or $\norm y_1 > 1$.
Su... | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_1 | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Trivial Norm/Division Ring"
] | [
"Definition:Contrapositive Statement",
"Definition:Trivial Norm/Division Ring/Nontrivial",
"Real Numbers form Totally Ordered Field",
"Definition:Norm/Division Ring",
"Properties of Norm on Division Ring/Norm of Inverse",
"Rule of Transposition",
"Category:Equivalence of Definitions of Equivalent Divisi... |
proofwiki-15333 | Syndrome is Zero iff Vector is Codeword | Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$
Let $G$ be a (standard) generator matrix for $C$.
Let $P$ be a standard parity check matrix for $C$.
Let $w \in \map V {n, p}$.
Then the syndrome of $w$ is zero {{iff}} $w$ is a codeword of $C$. | Let $G = \paren {\begin{array} {c|c} \mathbf I & \mathbf A \end{array} }$.
Let $c \in \map V {n, p}$.
Then, by definition of $G$, $c$ is a codeword of $C$ {{iff}} $c$ is of the form $u G$, where $u \in \map V {k, p}$.
Thus $c \in C$ {{iff}}:
{{begin-eqn}}
{{eqn | l = c
| r = u G
| c =
}}
{{eqn | r = u \par... | Let $C$ be a [[Definition:Linear Code|linear $\tuple {n, k}$-code]] whose [[Definition:Master Code|master code]] is $\map V {n, p}$
Let $G$ be a [[Definition:Standard Generator Matrix for Linear Code|(standard) generator matrix]] for $C$.
Let $P$ be a [[Definition:Standard Parity Check Matrix|standard parity check ma... | Let $G = \paren {\begin{array} {c|c} \mathbf I & \mathbf A \end{array} }$.
Let $c \in \map V {n, p}$.
Then, by definition of $G$, $c$ is a [[Definition:Codeword of Linear Code|codeword]] of $C$ {{iff}} $c$ is of the form $u G$, where $u \in \map V {k, p}$.
Thus $c \in C$ {{iff}}:
{{begin-eqn}}
{{eqn | l = c
|... | Syndrome is Zero iff Vector is Codeword | https://proofwiki.org/wiki/Syndrome_is_Zero_iff_Vector_is_Codeword | https://proofwiki.org/wiki/Syndrome_is_Zero_iff_Vector_is_Codeword | [
"Linear Codes"
] | [
"Definition:Linear Code",
"Definition:Linear Code/Master Code",
"Definition:Standard Generator Matrix for Linear Code",
"Definition:Standard Parity Check Matrix",
"Definition:Syndrome",
"Definition:Zero Codeword",
"Definition:Linear Code/Codeword"
] | [
"Definition:Linear Code/Codeword",
"Definition:Matrix",
"Definition:Syndrome",
"Definition:Syndrome",
"Definition:Zero Codeword",
"Definition:Concatenation of Matrices",
"Definition:Syndrome",
"Definition:Zero Codeword",
"Definition:Linear Code/Codeword"
] |
proofwiki-15334 | Condition for Vectors to have Same Syndrome | Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$
Let $G$ be a (standard) generator matrix for $C$.
Let $P$ be a standard parity check matrix for $C$.
Let $u, v \in \map V {n, p}$.
Then $u$ and $v$ have the same syndrome {{iff}} they are in the same coset of $C$. | Let $u, v \in \map V {n, p}$.
Let $\map S u$ denote the syndrome of $u$.
Then:
{{begin-eqn}}
{{eqn | l = \map S u
| r = \map S v
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = P u^\intercal
| r = P v^\intercal
| c = {{Defof|Syndrome}}
}}
{{eqn | ll= \leadstoandfrom
| l = P \paren {u^\in... | Let $C$ be a [[Definition:Linear Code|linear $\tuple {n, k}$-code]] whose [[Definition:Master Code|master code]] is $\map V {n, p}$
Let $G$ be a [[Definition:Standard Generator Matrix for Linear Code|(standard) generator matrix]] for $C$.
Let $P$ be a [[Definition:Standard Parity Check Matrix|standard parity check ma... | Let $u, v \in \map V {n, p}$.
Let $\map S u$ denote the [[Definition:Syndrome|syndrome]] of $u$.
Then:
{{begin-eqn}}
{{eqn | l = \map S u
| r = \map S v
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = P u^\intercal
| r = P v^\intercal
| c = {{Defof|Syndrome}}
}}
{{eqn | ll= \leadstoandfrom... | Condition for Vectors to have Same Syndrome | https://proofwiki.org/wiki/Condition_for_Vectors_to_have_Same_Syndrome | https://proofwiki.org/wiki/Condition_for_Vectors_to_have_Same_Syndrome | [
"Linear Codes"
] | [
"Definition:Linear Code",
"Definition:Linear Code/Master Code",
"Definition:Standard Generator Matrix for Linear Code",
"Definition:Standard Parity Check Matrix",
"Definition:Syndrome",
"Definition:Coset"
] | [
"Definition:Syndrome",
"Syndrome is Zero iff Vector is Codeword",
"Elements in Same Coset iff Product with Inverse in Subgroup"
] |
proofwiki-15335 | Euler's Equation/Independent of x | Let $y$ be a mapping.
Let $J$ be a functional such that:
:$\ds J \sqbrk y = \int_a^b \map F {y, y'} \rd x$
Then the corresponding Euler's Equation can be reduced to:
:$F - y' F_{y'} = C$
where $C$ is an arbitrary constant. | Assume that:
:$\ds J \sqbrk y = \int_a^b \map F {y, y'} \rd x$
Then:
{{begin-eqn}}
{{eqn| l = \delta J
| r = 0
}}
{{eqn| ll= \leadsto
| l = F_y - \dfrac \d {\d x} F_{y'}
| r = F_y - \paren {\dfrac {\d y} {\d x} \dfrac {\partial F_{y'} } {\partial y} + \dfrac {\d y'} {\d x} \dfrac {\partial F_{y'} } {\par... | Let $y$ be a [[Definition:Mapping|mapping]].
Let $J$ be a [[Definition:Real Functional|functional]] such that:
:$\ds J \sqbrk y = \int_a^b \map F {y, y'} \rd x$
Then the corresponding [[Definition:Euler's Equation for Vanishing Variation|Euler's Equation]] can be reduced to:
:$F - y' F_{y'} = C$
where $C$ is an ... | Assume that:
:$\ds J \sqbrk y = \int_a^b \map F {y, y'} \rd x$
Then:
{{begin-eqn}}
{{eqn| l = \delta J
| r = 0
}}
{{eqn| ll= \leadsto
| l = F_y - \dfrac \d {\d x} F_{y'}
| r = F_y - \paren {\dfrac {\d y} {\d x} \dfrac {\partial F_{y'} } {\partial y} + \dfrac {\d y'} {\d x} \dfrac {\partial F_{y'} } {\... | Euler's Equation/Independent of x | https://proofwiki.org/wiki/Euler's_Equation/Independent_of_x | https://proofwiki.org/wiki/Euler's_Equation/Independent_of_x | [
"Calculus of Variations"
] | [
"Definition:Mapping",
"Definition:Functional/Real",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Arbitrary Constant"
] | [
"Definition:Differential Equation",
"Definition:Primitive (Calculus)/Integration"
] |
proofwiki-15336 | Euler's Equation/Independent of y | Let $y$ be a mapping
Let $J$ be a functional such that
:$\ds J \sqbrk y = \int_a^b \map F {x,y'} \rd x$
Then the corresponding Euler's equation can be reduced to:
:$F_{y'} = C$
where $C$ is an arbitrary constant. | Assume that:
:$\ds J \sqbrk y = \int_a^b \map F {x,y'} \rd x$
Euler's equation for $J$ is:
:$\dfrac \d {\d x} F_{y'} = 0$
Integration yields:
:$F_{y'} = C$
{{qed}} | Let $y$ be a [[Definition:Mapping|mapping]]
Let $J$ be a [[Definition:Real Functional|functional]] such that
:$\ds J \sqbrk y = \int_a^b \map F {x,y'} \rd x$
Then the corresponding [[Definition:Euler's Equation for Vanishing Variation|Euler's equation]] can be reduced to:
:$F_{y'} = C$
where $C$ is an [[Definiti... | Assume that:
:$\ds J \sqbrk y = \int_a^b \map F {x,y'} \rd x$
[[Definition:Euler's Equation for Vanishing Variation|Euler's equation]] for $J$ is:
:$\dfrac \d {\d x} F_{y'} = 0$
[[Definition:Integration|Integration]] yields:
:$F_{y'} = C$
{{qed}} | Euler's Equation/Independent of y | https://proofwiki.org/wiki/Euler's_Equation/Independent_of_y | https://proofwiki.org/wiki/Euler's_Equation/Independent_of_y | [
"Calculus of Variations"
] | [
"Definition:Mapping",
"Definition:Functional/Real",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Arbitrary Constant"
] | [
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Primitive (Calculus)/Integration"
] |
proofwiki-15337 | Euler's Equation/Independent of y' | Let $y$ be a mapping.
Let $J$ a functional be such that
:$\ds J \sqbrk y = \int_a^b \map F {x,y} \rd x$
Then the corresponding Euler's Equation can be reduced to:
:$F_y = 0$
Furthermore, this is an algebraic equation. | Assume that:
:$\ds J \sqbrk y = \int_a^b \map F {x,y} \rd x$
Then Euler's Equation for $J$ is:
:$F_y = 0$
Since $F$ is independent of $y'$, the equation is algebraic.
{{qed}} | Let $y$ be a [[Definition:Mapping|mapping]].
Let $J$ a [[Definition:Real Functional|functional]] be such that
:$\ds J \sqbrk y = \int_a^b \map F {x,y} \rd x$
Then the corresponding [[Definition:Euler's Equation for Vanishing Variation|Euler's Equation]] can be reduced to:
:$F_y = 0$
Furthermore, this is an algeb... | Assume that:
:$\ds J \sqbrk y = \int_a^b \map F {x,y} \rd x$
Then [[Definition:Euler's Equation for Vanishing Variation|Euler's Equation]] for $J$ is:
:$F_y = 0$
Since $F$ is independent of $y'$, the [[Definition:Equation|equation]] is algebraic.
{{qed}} | Euler's Equation/Independent of y' | https://proofwiki.org/wiki/Euler's_Equation/Independent_of_y' | https://proofwiki.org/wiki/Euler's_Equation/Independent_of_y' | [
"Calculus of Variations"
] | [
"Definition:Mapping",
"Definition:Functional/Real",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Equation"
] | [
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Equation"
] |
proofwiki-15338 | Euler's Equation/Integrated wrt Length Element | Let $y$ be a real mapping belonging to $C^2$ differentiability class.
Assume that:
:$\ds J \sqbrk y = \int_a^b \map f {x, y, y'} \rd s$
where
:$\rd s = \sqrt {1 + y'^2} \rd x$
Then Euler's Equation can be reduced to:
:$f_y - f_x y' - f_{y'} y' y' ' - f \dfrac {y' '} {\paren {1 + y'^2}^{\frac 3 2} } = 0$ | Substitution of $\rd s$ into $J$ results in the following functional:
:$\ds J \sqbrk y = \int_a^b \map f {x, y, y'} \sqrt {1 + y'^2} \rd x$
We can consider this as a functional with the following effective $F$:
:$F = \map f {x, y, y'} \sqrt {1 + y'^2}$
Find Euler's Equation:
{{begin-eqn}}
{{eqn | l = F_y - \dfrac \d {\... | Let $y$ be a [[Definition:Real Function|real mapping]] belonging to $C^2$ [[Definition:Differentiability Class|differentiability class]].
Assume that:
:$\ds J \sqbrk y = \int_a^b \map f {x, y, y'} \rd s$
where
:$\rd s = \sqrt {1 + y'^2} \rd x$
Then [[Definition:Euler's Equation for Vanishing Variation|Euler's Equ... | Substitution of $\rd s$ into $J$ results in the following [[Definition:Real Functional|functional]]:
:$\ds J \sqbrk y = \int_a^b \map f {x, y, y'} \sqrt {1 + y'^2} \rd x$
We can consider this as a [[Definition:Real Functional|functional]] with the following effective $F$:
:$F = \map f {x, y, y'} \sqrt {1 + y'^2}$
F... | Euler's Equation/Integrated wrt Length Element | https://proofwiki.org/wiki/Euler's_Equation/Integrated_wrt_Length_Element | https://proofwiki.org/wiki/Euler's_Equation/Integrated_wrt_Length_Element | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Differentiability Class",
"Definition:Euler's Equation for Vanishing Variation"
] | [
"Definition:Functional/Real",
"Definition:Functional/Real",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Assumption",
"Definition:Euler's Equation for Vanishing Variation",
"Definition:Expression"
] |
proofwiki-15339 | Subset of Linear Code with Even Weight Codewords | Let $C$ be a linear code.
Let $C^+$ be the subset of $C$ consisting of all the codewords of $C$ which have even weight.
Then $C^+$ is a subgroup of $C$ such that either $C^+ = C$ or such that $\order {C^+} = \dfrac {\order C} 2$. | Note that the zero codeword is in $C^+$ as it has a weight of $0$ which is even.
Let $c$ and $d$ be of even weight, where $c$ and $d$ agree in $k$ ordinates.
Let $\map w c$ denote the weight of $c$.
Then:
{{begin-eqn}}
{{eqn | l = \map w {c + d}
| r = \map w c - k + \map w d - k
| c =
}}
{{eqn | r = \map w... | Let $C$ be a [[Definition:Linear Code|linear code]].
Let $C^+$ be the [[Definition:Subset|subset]] of $C$ consisting of all the [[Definition:Codeword of Linear Code|codewords]] of $C$ which have [[Definition:Even Integer|even]] [[Definition:Weight of Linear Codeword|weight]].
Then $C^+$ is a [[Definition:Subgroup|sub... | Note that the [[Definition:Zero Codeword|zero codeword]] is in $C^+$ as it has a [[Definition:Weight of Linear Codeword|weight]] of $0$ which is [[Definition:Even Integer|even]].
Let $c$ and $d$ be of [[Definition:Even Integer|even]] [[Definition:Weight of Linear Codeword|weight]], where $c$ and $d$ agree in $k$ ordin... | Subset of Linear Code with Even Weight Codewords | https://proofwiki.org/wiki/Subset_of_Linear_Code_with_Even_Weight_Codewords | https://proofwiki.org/wiki/Subset_of_Linear_Code_with_Even_Weight_Codewords | [
"Linear Codes"
] | [
"Definition:Linear Code",
"Definition:Subset",
"Definition:Linear Code/Codeword",
"Definition:Even Integer",
"Definition:Weight of Linear Codeword",
"Definition:Subgroup"
] | [
"Definition:Zero Codeword",
"Definition:Weight of Linear Codeword",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Weight of Linear Codeword",
"Definition:Weight of Linear Codeword",
"Definition:Even Integer",
"Definition:Vector/Linear Algebra",
"Definition:Inverse (Abstract Algeb... |
proofwiki-15340 | Minimal Smooth Surface of Revolution | Let $\map y x$ be a real mapping in 2-dimensional real Euclidean space.
Let $y$ pass through the points $\tuple {x_0, y_0}$ and $\tuple {x_1, y_1}$.
Consider a surface of revolution constructed by rotating $y$ around the $x$-axis.
Suppose this surface is smooth for any $x$ between $x_0$ and $x_1$.
Then its surface area... | The area functional of the surface of revolution is:
:$\ds A \sqbrk y = 2 \pi \int_{x_0}^{x_1} y \sqrt {1 + y'^2} \rd x$
The integrand does not depend on $x$.
By Euler's Equation:
:$F - y' F_{y'} = C$
that is:
:$y \sqrt {1 + y'^2} - \dfrac {y y'^2} {\sqrt {1 + y'^2} } = C$
which is equivalent to:
:$y = C \sqrt {1 + y'^... | Let $\map y x$ be a [[Definition:Real Function|real mapping]] in 2-dimensional [[Definition:Real Euclidean Space|real Euclidean space]].
Let $y$ pass through the [[Definition:Point|points]] $\tuple {x_0, y_0}$ and $\tuple {x_1, y_1}$.
Consider a [[Definition:Surface of Revolution|surface of revolution]] constructed b... | The [[Definition:Area|area]] [[Definition:Real Functional|functional]] of the [[Definition:Surface of Revolution|surface of revolution]] is:
:$\ds A \sqbrk y = 2 \pi \int_{x_0}^{x_1} y \sqrt {1 + y'^2} \rd x$
The [[Definition:Integrand|integrand]] does not depend on $x$.
By [[Euler's Equation/Independent of x|Euler'... | Minimal Smooth Surface of Revolution | https://proofwiki.org/wiki/Minimal_Smooth_Surface_of_Revolution | https://proofwiki.org/wiki/Minimal_Smooth_Surface_of_Revolution | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Euclidean Space/Real",
"Definition:Point",
"Definition:Surface of Revolution",
"Definition:Surface",
"Definition:Smooth Real Function",
"Definition:Surface",
"Definition:Area",
"Definition:Line/Curve",
"Definition:Area"
] | [
"Definition:Area",
"Definition:Functional/Real",
"Definition:Surface of Revolution",
"Definition:Integration/Integrand",
"Euler's Equation/Independent of x",
"Definition:Differential Equation",
"Definition:Primitive (Calculus)/Integration",
"Definition:Area",
"Definition:Real Function",
"Definitio... |
proofwiki-15341 | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2 | :$\forall x \in R: \norm x_1 = \norm x_2^\alpha$ | Since $\norm {x_0}_1 < 1$ then $\norm {x_0}_2 < 1$ and:
:$\log \norm {x_0}_1 < 0$
:$\log \norm {x_0}_2 < 0$
Hence $\alpha > 0$
Since $\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$: | :$\forall x \in R: \norm x_1 = \norm x_2^\alpha$ | Since $\norm {x_0}_1 < 1$ then $\norm {x_0}_2 < 1$ and:
:$\log \norm {x_0}_1 < 0$
:$\log \norm {x_0}_2 < 0$
Hence $\alpha > 0$
Since $\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$: | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2 | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [] | [] |
proofwiki-15342 | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.1 | Then:
:$\forall y \in R: \norm y_1 > 1 \iff \norm y_2 > 1$ | For $y \in R$ then:
{{begin-eqn}}
{{eqn | l = \norm y_1 > 1
| r = \dfrac 1 {\norm y_1 } < 1
| o = \leadstoandfrom
| c =
}}
{{eqn | r = \norm {y^{-1} }_1 < 1
| o = \leadstoandfrom
| c = Norm of Inverse in Division Ring
}}
{{eqn | r = \norm {y^{-1} }_2 < 1
| o = \leadstoandfrom
... | Then:
:$\forall y \in R: \norm y_1 > 1 \iff \norm y_2 > 1$ | For $y \in R$ then:
{{begin-eqn}}
{{eqn | l = \norm y_1 > 1
| r = \dfrac 1 {\norm y_1 } < 1
| o = \leadstoandfrom
| c =
}}
{{eqn | r = \norm {y^{-1} }_1 < 1
| o = \leadstoandfrom
| c = [[Norm of Inverse in Division Ring]]
}}
{{eqn | r = \norm {y^{-1} }_2 < 1
| o = \leadstoandfrom... | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2/Lemma_2.1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2/Lemma_2.1 | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [] | [
"Properties of Norm on Division Ring/Norm of Inverse",
"Properties of Norm on Division Ring/Norm of Inverse",
"Category:Equivalence of Definitions of Equivalent Division Ring Norms"
] |
proofwiki-15343 | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2 | Then:
:$\forall y \in R:\norm y_1 = 1 \iff \norm y_2 = 1$ | By assumption:
:$\forall y \in R:\norm y_1 \ge 1 \iff \norm y_2 \ge 1$
By Lemma 1:
:$\forall y \in R:\norm y_1 \le 1 \iff \norm y_2 \le 1$
Hence $\forall y \in R$:
{{begin-eqn}}
{{eqn | l = \norm y_1 = 1
| o = \leadstoandfrom
| r = \norm y_1 \le 1, \norm y_1 \ge 1
}}
{{eqn | o = \leadstoandfrom
| r =... | Then:
:$\forall y \in R:\norm y_1 = 1 \iff \norm y_2 = 1$ | By assumption:
:$\forall y \in R:\norm y_1 \ge 1 \iff \norm y_2 \ge 1$
By [[Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 1|Lemma 1]]:
:$\forall y \in R:\norm y_1 \le 1 \iff \norm y_2 \le 1$
Hence $\forall y \in R$:
{{begin-eqn}}
{{e... | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2/Lemma_2.2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2/Lemma_2.2 | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [] | [
"Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 1",
"Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 1",
"Category:Equivalence of Definitions of Equi... |
proofwiki-15344 | Symmetric Group on 3 Letters is Isomorphic to Dihedral Group D3 | Let $S_3$ denote the Symmetric Group on 3 Letters.
Let $D_3$ denote the dihedral group $D_3$.
Then $S_3$ is isomorphic to $D_3$. | Consider $S_3$ as presented by its Cayley table:
{{:Symmetric Group on 3 Letters/Cayley Table}}
Consider $D_3$ as presented by its group presentation:
{{:Group Presentation of Dihedral Group D3}}
and its Cayley table:
{{:Dihedral Group D3/Cayley Table}}
Let $\phi: S_3 \to D_3$ be specified as:
{{begin-eqn}}
{{eqn | l =... | Let $S_3$ denote the [[Symmetric Group on 3 Letters]].
Let $D_3$ denote the [[Definition:Dihedral Group D3|dihedral group $D_3$]].
Then $S_3$ is [[Definition:Isomorphism|isomorphic]] to $D_3$. | Consider $S_3$ as presented by its [[Symmetric Group on 3 Letters/Cayley Table|Cayley table]]:
{{:Symmetric Group on 3 Letters/Cayley Table}}
Consider $D_3$ as presented by its [[Group Presentation of Dihedral Group D3|group presentation]]:
{{:Group Presentation of Dihedral Group D3}}
and its [[Dihedral Group D3/Cayl... | Symmetric Group on 3 Letters is Isomorphic to Dihedral Group D3 | https://proofwiki.org/wiki/Symmetric_Group_on_3_Letters_is_Isomorphic_to_Dihedral_Group_D3 | https://proofwiki.org/wiki/Symmetric_Group_on_3_Letters_is_Isomorphic_to_Dihedral_Group_D3 | [
"Symmetric Group on 3 Letters",
"Dihedral Group D3",
"Examples of Group Isomorphisms"
] | [
"Symmetric Group on 3 Letters",
"Definition:Dihedral Group D3",
"Definition:Isomorphism"
] | [
"Symmetric Group on 3 Letters/Cayley Table",
"Dihedral Group D3/Group Presentation",
"Dihedral Group D3/Cayley Table"
] |
proofwiki-15345 | Homomorphism from Group of Cube Roots of Unity to Itself | Let $\struct {U_3, \times}$ denote the multiplicative group of the complex cube roots of unity.
Here, $U_3 = \set {1, \omega, \omega^2}$ where $\omega = e^{2 i \pi / 3}$.
Let $\phi: U_3 \to U_3$ be defined as:
:$\forall z \in U_3: \map \phi z = \begin{cases} 1 & : z = 1 \\ \omega^2 & : z = \omega \\ \omega & : z = \ome... | It is noted that
:$\paren {\omega^2}^2 = \omega$
and so $\phi$ is the square function.
By Roots of Unity under Multiplication form Cyclic Group and Cyclic Group is Abelian, $U_3$ is abelian.
Thus for all $a, b \in U_3$:
{{begin-eqn}}
{{eqn | l = \map \phi a \map \phi b
| r = a^2 b^2
}}
{{eqn | r = a b a b
|... | Let $\struct {U_3, \times}$ denote the [[Definition:Multiplicative Group of Complex Roots of Unity|multiplicative group of the complex cube roots of unity]].
Here, $U_3 = \set {1, \omega, \omega^2}$ where $\omega = e^{2 i \pi / 3}$.
Let $\phi: U_3 \to U_3$ be defined as:
:$\forall z \in U_3: \map \phi z = \begin{ca... | It is noted that
:$\paren {\omega^2}^2 = \omega$
and so $\phi$ is the [[Definition:Square Function|square function]].
By [[Roots of Unity under Multiplication form Cyclic Group]] and [[Cyclic Group is Abelian]], $U_3$ is [[Definition:Abelian Group|abelian]].
Thus for all $a, b \in U_3$:
{{begin-eqn}}
{{eqn | l = \... | Homomorphism from Group of Cube Roots of Unity to Itself | https://proofwiki.org/wiki/Homomorphism_from_Group_of_Cube_Roots_of_Unity_to_Itself | https://proofwiki.org/wiki/Homomorphism_from_Group_of_Cube_Roots_of_Unity_to_Itself | [
"Multiplicative Groups of Complex Roots of Unity",
"Cyclic Group of Order 3"
] | [
"Definition:Multiplicative Group of Complex Roots of Unity",
"Definition:Group Homomorphism"
] | [
"Definition:Square/Function",
"Roots of Unity under Multiplication form Cyclic Group",
"Cyclic Group is Abelian",
"Definition:Abelian Group",
"Definition:Group Homomorphism"
] |
proofwiki-15346 | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.3 | :$\alpha = \beta$ | Because $x, y \in R \setminus 0_R$:
:$\norm x_1 , \norm y_1, \norm x_2 , \norm y_2 > 0$.
Because $\norm{x}_1 , \norm {y}_1 \ne 1$, by Lemma 2:
:$\norm x_2 , \norm y_2 \ne 1$.
Hence:
:$\log \norm x_1 , \log \norm y_1, \log \norm x_2, \log \norm y_2 \ne 0$
and $\alpha, \beta$ are well-defined.
Let $r = \dfrac n m \in \Q... | :$\alpha = \beta$ | Because $x, y \in R \setminus 0_R$:
:$\norm x_1 , \norm y_1, \norm x_2 , \norm y_2 > 0$.
Because $\norm{x}_1 , \norm {y}_1 \ne 1$, by [[Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2|Lemma 2]]:
:$\norm x_2 , \norm y_2 \ne 1$.... | Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2/Lemma_2.3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Open_Unit_Ball_Equivalent_implies_Norm_is_Power_of_Other_Norm/Lemma_2/Lemma_2.3 | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [] | [
"Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2",
"Definition:Rational Number",
"Definition:Integer",
"Properties of Norm on Division Ring/Norm of Inverse",
"Definition:Norm/Division Ring",
"Definition:Norm/Divis... |
proofwiki-15347 | Product with Inverse on Homomorphic Image is Group Homomorphism | Let $G$ be a group.
Let $H$ be an abelian group.
Let $\theta: G \to H$ be a (group) homomorphism.
Let $\phi: G \times G \to H$ be the mapping defined as:
:$\forall \tuple {g_1, g_2} \in G \times G: \map \phi {g_1, g_2} = \map \theta {g_1} \map \theta {g_2}^{-1}$
Then $\phi$ is a homomorphism. | First note that from Group Homomorphism Preserves Inverses:
:$\map \theta {g_2}^{-1} = \paren {\map \theta {g_2} }^{-1} = \map \theta { {g_2}^{-1} }$
and so $\map \theta {g_1} \map \theta {g_2}^{-1}$ is not ambiguous:
:$\map \theta {g_1} \map \theta {g_2}^{-1} = \map \theta {g_1} \paren {\map \theta {g_2} }^{-1} = \map... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be an [[Definition:Abelian Group|abelian group]].
Let $\theta: G \to H$ be a [[Definition:Group Homomorphism|(group) homomorphism]].
Let $\phi: G \times G \to H$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall \tuple {g_1, g_2} \in G \times G: \map \phi ... | First note that from [[Group Homomorphism Preserves Inverses]]:
:$\map \theta {g_2}^{-1} = \paren {\map \theta {g_2} }^{-1} = \map \theta { {g_2}^{-1} }$
and so $\map \theta {g_1} \map \theta {g_2}^{-1}$ is not [[Definition:Ambiguous|ambiguous]]:
:$\map \theta {g_1} \map \theta {g_2}^{-1} = \map \theta {g_1} \paren ... | Product with Inverse on Homomorphic Image is Group Homomorphism | https://proofwiki.org/wiki/Product_with_Inverse_on_Homomorphic_Image_is_Group_Homomorphism | https://proofwiki.org/wiki/Product_with_Inverse_on_Homomorphic_Image_is_Group_Homomorphism | [
"Group Homomorphisms",
"Group Direct Products",
"Product with Inverse on Homomorphic Image is Group Homomorphism"
] | [
"Definition:Group",
"Definition:Abelian Group",
"Definition:Group Homomorphism",
"Definition:Mapping",
"Definition:Group Homomorphism"
] | [
"Group Homomorphism Preserves Inverses",
"Definition:Ambiguity",
"External Direct Product of Groups is Group",
"Definition:Group",
"Group Homomorphism Preserves Inverses",
"Inverse of Group Product",
"Definition:Abelian Group",
"Definition:Group Homomorphism"
] |
proofwiki-15348 | Max Operation is Associative | The Max operation is associative:
: $\map \max {\map \max {x, y}, z} = \map \max {x, \max \paren{y, z}}$
Thus we are justified in writing $\map \max {x, y, z}$. | To simplify our notation:
: Let $\map \max {x, y}$ be (temporarily) denoted $x \overline \wedge y$
There are the following cases to consider:
:$(1): \quad x \le y \le z$
:$(2): \quad x \le z \le y$
:$(3): \quad y \le x \le z$
:$(4): \quad y \le z \le x$
:$(5): \quad z \le x \le y$
:$(6): \quad z \le y \le x$
Taking eac... | The [[Definition:Max Operation|Max]] operation is [[Definition:Associative Operation|associative]]:
: $\map \max {\map \max {x, y}, z} = \map \max {x, \max \paren{y, z}}$
Thus we are justified in writing $\map \max {x, y, z}$. | To simplify our notation:
: Let $\map \max {x, y}$ be (temporarily) denoted $x \overline \wedge y$
There are the following cases to consider:
:$(1): \quad x \le y \le z$
:$(2): \quad x \le z \le y$
:$(3): \quad y \le x \le z$
:$(4): \quad y \le z \le x$
:$(5): \quad z \le x \le y$
:$(6): \quad z \le y \le x$
Taking... | Max Operation is Associative | https://proofwiki.org/wiki/Max_Operation_is_Associative | https://proofwiki.org/wiki/Max_Operation_is_Associative | [
"Max Operation",
"Examples of Associative Operations"
] | [
"Definition:Max Operation",
"Definition:Associative Operation"
] | [] |
proofwiki-15349 | Min Operation is Associative | The min operation is associative:
:$\map \min {\map \min {x, y}, z} = \map \min {x, \map \min {y, z} }$
Thus we are justified in writing $\map \min {x, y, z}$. | To simplify our notation:
:Let $\map \min {x, y}$ be (temporarily) denoted $x \underline \vee y$.
There are the following cases to consider:
:$(1): \quad x \le y \le z$
:$(2): \quad x \le z \le y$
:$(3): \quad y \le x \le z$
:$(4): \quad y \le z \le x$
:$(5): \quad z \le x \le y$
:$(6): \quad z \le y \le x$
Taking each... | The [[Definition:Min Operation|min operation]] is [[Definition:Associative Operation|associative]]:
:$\map \min {\map \min {x, y}, z} = \map \min {x, \map \min {y, z} }$
Thus we are justified in writing $\map \min {x, y, z}$. | To simplify our notation:
:Let $\map \min {x, y}$ be (temporarily) denoted $x \underline \vee y$.
There are the following cases to consider:
:$(1): \quad x \le y \le z$
:$(2): \quad x \le z \le y$
:$(3): \quad y \le x \le z$
:$(4): \quad y \le z \le x$
:$(5): \quad z \le x \le y$
:$(6): \quad z \le y \le x$
Taking ... | Min Operation is Associative | https://proofwiki.org/wiki/Min_Operation_is_Associative | https://proofwiki.org/wiki/Min_Operation_is_Associative | [
"Min Operation",
"Examples of Associative Operations"
] | [
"Definition:Min Operation",
"Definition:Associative Operation"
] | [] |
proofwiki-15350 | Theoretical Justification for Cycle Notation | Let $\N_k$ be used to denote the initial segment of natural numbers:
:$\N_k = \closedint 1 k = \set {1, 2, 3, \ldots, k}$
Let $\rho: \N_n \to \N_n$ be a permutation of $n$ letters.
Let $i \in \N_n$.
Let $k$ be the smallest (strictly) positive integer for which $\map {\rho^k} i$ is in the set:
:$\set {i, \map \rho i, \m... | {{AimForCont}} $\map {\rho^k} i = \map {\rho^r} i$ for some $r > 0$.
As $\rho$ has an inverse in $S_n$:
:$\map {\rho^{k - r} } i = i$
This contradicts the definition of $k$, because $k - r < k$
Thus:
:$r = 0$
The result follows.
{{qed}} | Let $\N_k$ be used to denote the [[Definition:Initial Segment of Natural Numbers|initial segment of natural numbers]]:
:$\N_k = \closedint 1 k = \set {1, 2, 3, \ldots, k}$
Let $\rho: \N_n \to \N_n$ be a [[Definition:Permutation on n Letters|permutation of $n$ letters]].
Let $i \in \N_n$.
Let $k$ be the smallest [[D... | {{AimForCont}} $\map {\rho^k} i = \map {\rho^r} i$ for some $r > 0$.
As $\rho$ has an [[Definition:Inverse Element|inverse]] in $S_n$:
:$\map {\rho^{k - r} } i = i$
This [[Definition:Contradiction|contradicts]] the definition of $k$, because $k - r < k$
Thus:
:$r = 0$
The result follows.
{{qed}} | Theoretical Justification for Cycle Notation | https://proofwiki.org/wiki/Theoretical_Justification_for_Cycle_Notation | https://proofwiki.org/wiki/Theoretical_Justification_for_Cycle_Notation | [
"Permutations"
] | [
"Definition:Initial Segment of Natural Numbers",
"Definition:Permutation on n Letters",
"Definition:Strictly Positive/Integer",
"Definition:Set"
] | [
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Contradiction"
] |
proofwiki-15351 | Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Topologically Equivalent | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:$\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
Then $d_1$ and $d_2$ are topologically equivalent metrics. | Let $x \in R$ and $\epsilon \in \R_{\gt 0}$
Then for $y \in R$:
{{begin-eqn}}
{{eqn | l = \norm {y - x}_1 < \epsilon
| o = \leadstoandfrom
| r = \norm {y - x}_2^\alpha < \epsilon
}}
{{eqn | o = \leadstoandfrom
| r = \norm {y - x}_2 < \epsilon^{1 / \alpha}
}}
{{end-eqn}}
Hence:
:$\map {B^1_\epsilon} x ... | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:$\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
Then $d_1$ and $d_2$ are [[Definition:Topologically Equivalent Metrics|topologically equivalent metrics]]. | Let $x \in R$ and $\epsilon \in \R_{\gt 0}$
Then for $y \in R$:
{{begin-eqn}}
{{eqn | l = \norm {y - x}_1 < \epsilon
| o = \leadstoandfrom
| r = \norm {y - x}_2^\alpha < \epsilon
}}
{{eqn | o = \leadstoandfrom
| r = \norm {y - x}_2 < \epsilon^{1 / \alpha}
}}
{{end-eqn}}
Hence:
:$\map {B^1_\epsilon} ... | Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Topologically Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Norm_is_Power_of_Other_Norm_implies_Topologically_Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Norm_is_Power_of_Other_Norm_implies_Topologically_Equivalent | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Topologically Equivalent Metrics"
] | [
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Set/Metric ... |
proofwiki-15352 | Equivalence of Definitions of Equivalent Division Ring Norms/Cauchy Sequence Equivalent implies Open Unit Ball Equivalent | Let $R$ be a division ring.
Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be norms on $R$.
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:for all sequences $\sequence {x_n}$ in $R$: $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_1$ {{iff}}... | The contrapositive is proved.
Let there exist $x \in R$ such that $\norm x_1 < 1$ and $\norm x_2 \ge 1$.
Let $\sequence {x_n}$ be the sequence defined by: $\forall n: x_n = x^n$.
By Sequence of Powers of Number less than One in Normed Division Ring then $\sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_1$.
B... | Let $R$ be a [[Definition:Division Ring|division ring]].
Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be [[Definition:Norm on Division Ring|norms]] on $R$.
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:for all sequences $\sequence {x_n}$ in $R$: $\seq... | The [[Definition:Contrapositive|contrapositive]] is proved.
Let there exist $x \in R$ such that $\norm x_1 < 1$ and $\norm x_2 \ge 1$.
Let $\sequence {x_n}$ be the [[Definition:Sequence|sequence]] defined by: $\forall n: x_n = x^n$.
By [[Sequence of Powers of Number less than One/Normed Division Ring|Sequence of Po... | Equivalence of Definitions of Equivalent Division Ring Norms/Cauchy Sequence Equivalent implies Open Unit Ball Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Cauchy_Sequence_Equivalent_implies_Open_Unit_Ball_Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Cauchy_Sequence_Equivalent_implies_Open_Unit_Ball_Equivalent | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Division Ring",
"Definition:Norm/Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Contrapositive Statement",
"Definition:Sequence",
"Sequence of Powers of Number less than One/Normed Division Ring",
"Definition:Null Sequence/Normed Division Ring",
"Convergent Sequence is Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:... |
proofwiki-15353 | Equal Order Elements may not be Conjugate | Let $G$ be a group
Let $x, y \in G$ be elements of $G$ such that:
:$\order x = \order y$
where $\order x$ denotes the order of $x$.
Then it is not necessarily the case that $x$ and $y$ are conjugates. | Consider the dihedral group $D_4$, whose group presentation is:
{{:Group Presentation of Dihedral Group D4}}
We have that:
:$\order {a^2} = 2$
and:
:$\order b = 2$
but $a^2$ and $b$ are not conjugate to each other.
{{qed}} | Let $G$ be a [[Definition:Group|group]]
Let $x, y \in G$ be [[Definition:Element|elements]] of $G$ such that:
:$\order x = \order y$
where $\order x$ denotes the [[Definition:Order of Group Element|order]] of $x$.
Then it is not necessarily the case that $x$ and $y$ are [[Definition:Conjugate of Group Element|conjug... | Consider the [[Definition:Dihedral Group D4|dihedral group $D_4$]], whose [[Group Presentation of Dihedral Group D4|group presentation]] is:
{{:Group Presentation of Dihedral Group D4}}
We have that:
:$\order {a^2} = 2$
and:
:$\order b = 2$
but $a^2$ and $b$ are not [[Definition:Conjugate of Group Element|conjugate]... | Equal Order Elements may not be Conjugate | https://proofwiki.org/wiki/Equal_Order_Elements_may_not_be_Conjugate | https://proofwiki.org/wiki/Equal_Order_Elements_may_not_be_Conjugate | [
"Order of Group Elements",
"Conjugacy"
] | [
"Definition:Group",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Conjugate (Group Theory)/Element"
] | [
"Definition:Dihedral Group D4",
"Dihedral Group D4/Group Presentation",
"Definition:Conjugate (Group Theory)/Element"
] |
proofwiki-15354 | Group Action of Symmetric Group on Complex Vector Space | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $V$ denote a vector space over the complex numbers $\C$.
Let $V$ have a basis:
:$\BB := \set {v_1, v_2, \ldots, v_n}$
Let $*: S_n \times V \to V$ be a group action of $S_n$ on $V$ defined as:
:$\forall \tuple ... | Let $\rho, \sigma \in S_n$.
Let $v = \ds \sum_{k \mathop = 1}^n \lambda_k v_k$.
We have:
{{begin-eqn}}
{{eqn | l = \rho * \paren {\sigma * v}
| r = \rho * \paren {\sigma * \sum_{k \mathop = 1}^n \lambda_k v_k}
| c = Definition of $v$
}}
{{eqn | r = \rho * \sum_{k \mathop = 1}^n \lambda_k v_{\map \sigma k}
... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $V$ denote a [[Definition:Vector Space|vector space]] over the [[Definition:Complex Number|complex numbers $\C$]].
Let $V... | Let $\rho, \sigma \in S_n$.
Let $v = \ds \sum_{k \mathop = 1}^n \lambda_k v_k$.
We have:
{{begin-eqn}}
{{eqn | l = \rho * \paren {\sigma * v}
| r = \rho * \paren {\sigma * \sum_{k \mathop = 1}^n \lambda_k v_k}
| c = Definition of $v$
}}
{{eqn | r = \rho * \sum_{k \mathop = 1}^n \lambda_k v_{\map \sigma ... | Group Action of Symmetric Group on Complex Vector Space | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Complex_Vector_Space | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Complex_Vector_Space | [
"Examples of Group Actions",
"Group Action of Symmetric Group on Complex Vector Space",
"Symmetric Groups"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Symmetric Group/n Letters",
"Definition:Vector Space",
"Definition:Complex Number",
"Definition:Basis of Vector Space",
"Definition:Group Action",
"Definition:Group Action"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-15355 | Group Action of Symmetric Group on Complex Vector Space/Orbit | The orbit of an element $v \in V$ is:
:$\ds \Orb v = \set {w \in V: \exists \rho \in S_n: w = \sum_{k \mathop = 1}^n \lambda_k v_{\map \rho k} }$ | By definition:
{{begin-eqn}}
{{eqn | l = \Orb v
| r = \set {w \in V: \exists \rho \in S_n: w = \rho * v}
| c = {{Defof|Orbit (Group Theory)|Orbit}}
}}
{{eqn | r = \set {w \in V: \exists \rho \in S_n: w = \rho * \sum_{k \mathop = 1}^n \lambda_k v_k}
| c = Definition of $v$
}}
{{eqn | r = \set {w \in V:... | The [[Definition:Orbit (Group Theory)|orbit]] of an [[Definition:Element|element]] $v \in V$ is:
:$\ds \Orb v = \set {w \in V: \exists \rho \in S_n: w = \sum_{k \mathop = 1}^n \lambda_k v_{\map \rho k} }$ | By definition:
{{begin-eqn}}
{{eqn | l = \Orb v
| r = \set {w \in V: \exists \rho \in S_n: w = \rho * v}
| c = {{Defof|Orbit (Group Theory)|Orbit}}
}}
{{eqn | r = \set {w \in V: \exists \rho \in S_n: w = \rho * \sum_{k \mathop = 1}^n \lambda_k v_k}
| c = Definition of $v$
}}
{{eqn | r = \set {w \in V... | Group Action of Symmetric Group on Complex Vector Space/Orbit | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Complex_Vector_Space/Orbit | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Complex_Vector_Space/Orbit | [
"Group Action of Symmetric Group on Complex Vector Space"
] | [
"Definition:Orbit (Group Theory)",
"Definition:Element"
] | [] |
proofwiki-15356 | Group Action of Symmetric Group on Complex Vector Space/Stabilizer | The stabilizer of an element $v \in V$ is:
:$\ds \Stab v = \set {\rho \in S_n: \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_{\map \rho k} v_k}$ | By definition:
{{begin-eqn}}
{{eqn | l = \Stab v
| r = \set {\rho \in S_n: \rho * v = v}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {\rho \in S_n: \rho * \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_k v_k}
| c = Definition of $v$
}}
{{eqn | r = \set {\rho \in S_n: \sum_{k ... | The [[Definition:Stabilizer|stabilizer]] of an [[Definition:Element|element]] $v \in V$ is:
:$\ds \Stab v = \set {\rho \in S_n: \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_{\map \rho k} v_k}$ | By definition:
{{begin-eqn}}
{{eqn | l = \Stab v
| r = \set {\rho \in S_n: \rho * v = v}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {\rho \in S_n: \rho * \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_k v_k}
| c = Definition of $v$
}}
{{eqn | r = \set {\rho \in S_n: \sum_{k... | Group Action of Symmetric Group on Complex Vector Space/Stabilizer | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Complex_Vector_Space/Stabilizer | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Complex_Vector_Space/Stabilizer | [
"Stabilizers",
"Group Action of Symmetric Group on Complex Vector Space"
] | [
"Definition:Stabilizer",
"Definition:Element"
] | [] |
proofwiki-15357 | Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Cauchy Sequence Equivalent | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:$\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
Then for all sequences $\sequence {x_n}$ in $R$:
:$\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_1$ {{iff}} $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, ... | Let $\sequence {x_n}$ be a Cauchy sequence in $\norm {\, \cdot \,}_1$.
Let $\epsilon > 0$ be given.
Since $\sequence {x_n}$ is a Cauchy sequence then:
:$\exists N \in \N: \forall n,m \ge N: \norm {x_n - x_m}_1 < \epsilon^\alpha$
Then:
:$\exists N \in \N: \forall n,m \ge N: \norm {x_n - x_m}_2^\alpha < \epsilon^\alpha$
... | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
:$\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
Then for all sequences $\sequence {x_n}$ in $R$:
:$\sequence {x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in $\norm {\, \cdot \,}_1$ {{if... | Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in $\norm {\, \cdot \,}_1$.
Let $\epsilon > 0$ be given.
Since $\sequence {x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] then:
:$\exists N \in \N: \forall n,m \ge N: \norm {x_n - x_m}... | Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Cauchy Sequence Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Norm_is_Power_of_Other_Norm_implies_Cauchy_Sequence_Equivalent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Equivalent_Division_Ring_Norms/Norm_is_Power_of_Other_Norm_implies_Cauchy_Sequence_Equivalent | [
"Equivalence of Definitions of Equivalent Division Ring Norms"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
... |
proofwiki-15358 | Conjugacy Classes of Symmetric Group | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $S_n$ denote the symmetric group on $n$ letters.
The conjugacy classes of $S_n$ are determined entirely by the cycle type.
That is, the conjugacy class $\conjclass x$ of an element $x$ of $S_n$ consists of all the elements of $S_n$ whose cycle type is the same a... | Let $\sigma \in S_n$ have cycle type $\tuple {k_1, k_2, \ldots, k_n}$.
Let $\rho$ be conjugate to $\sigma$
From Conjugate Permutations have Same Cycle Type, $\rho$ has the same cycle type $\tuple {k_1, k_2, \ldots, k_n}$ as $\sigma$.
That is, all the elements of the same conjugacy class have the same cycle type.
{{qed|... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
The [[Definition:Conjugacy Class|conjugacy classes]] of $S_n$ are determined entirely by the [[Definition:Cycle Type|cycle ty... | Let $\sigma \in S_n$ have [[Definition:Cycle Type|cycle type]] $\tuple {k_1, k_2, \ldots, k_n}$.
Let $\rho$ be [[Definition:Conjugate of Group Element|conjugate]] to $\sigma$
From [[Conjugate Permutations have Same Cycle Type]], $\rho$ has the same [[Definition:Cycle Type|cycle type]] $\tuple {k_1, k_2, \ldots, k_n}$... | Conjugacy Classes of Symmetric Group | https://proofwiki.org/wiki/Conjugacy_Classes_of_Symmetric_Group | https://proofwiki.org/wiki/Conjugacy_Classes_of_Symmetric_Group | [
"Symmetric Groups",
"Conjugacy Classes"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Symmetric Group/n Letters",
"Definition:Conjugacy Class",
"Definition:Cycle Type",
"Definition:Conjugacy Class",
"Definition:Element",
"Definition:Element",
"Definition:Cycle Type",
"Definition:Cycle Type"
] | [
"Definition:Cycle Type",
"Definition:Conjugate (Group Theory)/Element",
"Conjugate Permutations have Same Cycle Type",
"Definition:Cycle Type",
"Definition:Element",
"Definition:Conjugacy Class",
"Definition:Cycle Type",
"Definition:Cycle Type",
"Definition:Conjugacy Class",
"Existence and Uniquen... |
proofwiki-15359 | Identity of Group is in Center | Let $G$ be a group.
Let $e$ be the identity of $G$.
Then $e$ is in the center of $G$:
:$e \in \map Z G$ | From Center is Intersection of Centralizers:
:$\ds \map Z G = \bigcap_{g \mathop \in G} \map {C_G} g$
where $\map {C_G} g$ denotes the centralizer of $g$.
From Centralizer of Group Element is Subgroup, each of $\map {C_G} g$ is a subgroup of $G$.
From Identity of Subgroup:
:$\forall g \in G: e \in \map {C_G} g$
Hence b... | Let $G$ be a [[Definition:Group|group]].
Let $e$ be the [[Definition:Identity Element|identity]] of $G$.
Then $e$ is in the [[Definition:Center of Group|center]] of $G$:
:$e \in \map Z G$ | From [[Center is Intersection of Centralizers]]:
:$\ds \map Z G = \bigcap_{g \mathop \in G} \map {C_G} g$
where $\map {C_G} g$ denotes the [[Definition:Centralizer of Group Element|centralizer]] of $g$.
From [[Centralizer of Group Element is Subgroup]], each of $\map {C_G} g$ is a [[Definition:Subgroup|subgroup]] of... | Identity of Group is in Center | https://proofwiki.org/wiki/Identity_of_Group_is_in_Center | https://proofwiki.org/wiki/Identity_of_Group_is_in_Center | [
"Centers of Groups",
"Identity Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Center (Abstract Algebra)/Group"
] | [
"Center is Intersection of Centralizers",
"Definition:Centralizer/Group Element",
"Centralizer of Group Element is Subgroup",
"Definition:Subgroup",
"Identity of Subgroup",
"Definition:Set Intersection",
"Category:Centers of Groups",
"Category:Identity Elements"
] |
proofwiki-15360 | Identity of Group is in Singleton Conjugacy Class | Let $G$ be a group.
Let $e$ be the identity of $G$.
Then $e$ is in its own singleton conjugacy class:
:$\conjclass e = \set e$ | From Identity of Group is in Center:
:$e \in \map Z G$
where $\map Z G$ is the center of $G$.
From Conjugacy Class of Element of Center is Singleton:
:$\conjclass e = \set e$
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $e$ be the [[Definition:Identity Element|identity]] of $G$.
Then $e$ is in its own [[Definition:Singleton|singleton]] [[Definition:Conjugacy Class|conjugacy class]]:
:$\conjclass e = \set e$ | From [[Identity of Group is in Center]]:
:$e \in \map Z G$
where $\map Z G$ is the [[Definition:Center of Group|center]] of $G$.
From [[Conjugacy Class of Element of Center is Singleton]]:
:$\conjclass e = \set e$
{{qed}} | Identity of Group is in Singleton Conjugacy Class | https://proofwiki.org/wiki/Identity_of_Group_is_in_Singleton_Conjugacy_Class | https://proofwiki.org/wiki/Identity_of_Group_is_in_Singleton_Conjugacy_Class | [
"Identity Elements",
"Conjugacy Classes"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Singleton",
"Definition:Conjugacy Class"
] | [
"Identity of Group is in Center",
"Definition:Center (Abstract Algebra)/Group",
"Conjugacy Class of Element of Center is Singleton"
] |
proofwiki-15361 | Finite Group with 2 Conjugacy Classes has 2 Elements | Let $G$ be a finite group.
Let $G$ have exactly $2$ conjugacy classes.
Then $G$ has exactly $2$ elements. | Let $G$ be of order $n$.
Let $G$ have exactly $2$ conjugacy classes.
Let $x \in G$ such that $x \ne e$.
Let $\conjclass x$ denote the conjugacy class of $x$.
From Identity of Group is in Singleton Conjugacy Class:
$\conjclass e = \set e$
where $\conjclass e$ denotes the conjugacy class of $e$
The other elements of $G$ ... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $G$ have exactly $2$ [[Definition:Conjugacy Class|conjugacy classes]].
Then $G$ has exactly $2$ [[Definition:Element|elements]]. | Let $G$ be of [[Definition:Order of Group|order]] $n$.
Let $G$ have exactly $2$ [[Definition:Conjugacy Class|conjugacy classes]].
Let $x \in G$ such that $x \ne e$.
Let $\conjclass x$ denote the [[Definition:Conjugacy Class|conjugacy class]] of $x$.
From [[Identity of Group is in Singleton Conjugacy Class]]:
$\con... | Finite Group with 2 Conjugacy Classes has 2 Elements | https://proofwiki.org/wiki/Finite_Group_with_2_Conjugacy_Classes_has_2_Elements | https://proofwiki.org/wiki/Finite_Group_with_2_Conjugacy_Classes_has_2_Elements | [
"Conjugacy Classes"
] | [
"Definition:Finite Group",
"Definition:Conjugacy Class",
"Definition:Element"
] | [
"Definition:Order of Structure",
"Definition:Conjugacy Class",
"Definition:Conjugacy Class",
"Identity of Group is in Singleton Conjugacy Class",
"Definition:Conjugacy Class",
"Definition:Element",
"Definition:Cardinality",
"Definition:Set",
"Size of Conjugacy Class is Index of Normalizer",
"Lagra... |
proofwiki-15362 | Group of Order 15 has Cyclic Subgroups of Order 3 and Order 5 | Let $G$ be a group whose order is $15$.
Then $G$ has
:a cyclic subgroup of order $3$
and:
:a cyclic subgroup of order $5$. | Let $G$ be a group of order $15$.
We have that $15 = 3 \times 5$.
Thus from the First Sylow Theorem:
:$G$ has at least one subgroup $H_3$ of order $3$
and:
:$G$ has at least one subgroup $H_5$ of order $5$.
From Prime Group is Cyclic, all such subgroups of order $3$ and order $5$ are cyclic.
{{Qed}} | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $15$.
Then $G$ has
:a [[Definition:Cyclic Group|cyclic]] [[Definition:Subgroup|subgroup]] of [[Definition:Order of Group|order]] $3$
and:
:a [[Definition:Cyclic Group|cyclic]] [[Definition:Subgroup|subgroup]] of [[Definition:Order of ... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $15$.
We have that $15 = 3 \times 5$.
Thus from the [[First Sylow Theorem]]:
:$G$ has at least one [[Definition:Subgroup|subgroup]] $H_3$ of [[Definition:Order of Group|order]] $3$
and:
:$G$ has at least one [[Definition:Subgroup|subgroup]... | Group of Order 15 has Cyclic Subgroups of Order 3 and Order 5 | https://proofwiki.org/wiki/Group_of_Order_15_has_Cyclic_Subgroups_of_Order_3_and_Order_5 | https://proofwiki.org/wiki/Group_of_Order_15_has_Cyclic_Subgroups_of_Order_3_and_Order_5 | [
"Groups of Order 15"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Subgroup",
"Definition:Order of Structure"
] | [
"Definition:Group",
"Definition:Order of Structure",
"First Sylow Theorem",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Subgroup",
"Definition:Order of Structure",
"Prime Group is Cyclic",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Order of Struct... |
proofwiki-15363 | Number of Sylow p-Subgroups in Group of Order 15 | Let $G$ be a group whose order is $15$.
Then:
: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 $\set {1, 6, 11, \ldots}$ | Let $G$ be a group of order $15$.
From the Fourth Sylow Theorem:
:the number of Sylow $p$-subgroups is equivalent to $1 \pmod p$
We have that $15 = 3 \times 5$.
Hence the result.
{{Qed}} | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $15$.
Then:
: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... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $15$.
From the [[Fourth Sylow Theorem]]:
:the number of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] is equivalent to $1 \pmod p$
We have that $15 = 3 \times 5$.
Hence the result.
{{Qed}} | Number of Sylow p-Subgroups in Group of Order 15 | https://proofwiki.org/wiki/Number_of_Sylow_p-Subgroups_in_Group_of_Order_15 | https://proofwiki.org/wiki/Number_of_Sylow_p-Subgroups_in_Group_of_Order_15 | [
"Groups of Order 15",
"Sylow p-Subgroups"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup",
"Definition:Set",
"Definition:Sylow p-Subgroup",
"Definition:Set"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Fourth Sylow Theorem",
"Definition:Sylow p-Subgroup"
] |
proofwiki-15364 | Direct Product of Sylow p-Subgroups is Sylow p-Subgroup | Let $G_1$ and $G_2$ be groups.
Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively.
Let $H_1$ be a Sylow $p$-subgroup of $G_1$.
Let $H_2$ be a Sylow $p$-subgroup of $G_2$.
Then $H_1 \times H_2$ is a Sylow $p$-subgroup of $G_1 \times G_2$. | By definition of Sylow $p$-subgroup:
:$\order {H_1} = p^r$, where $p^r$ is the highest power of $p$ which is a divisor of $\order {G_1}$.
:$\order {H_2} = p^s$, where $p^s$ is the highest power of $p$ which is a divisor of $\order {G_2}$.
We have that:
:$\order {H_1 \times H_2} = p^{r + s}$
We also have that $p^{r + s}... | Let $G_1$ and $G_2$ be [[Definition:Group|groups]].
Let $H_1$ and $H_2$ be [[Definition:Subgroup|subgroups]] of $G_1$ and $G_2$ respectively.
Let $H_1$ be a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G_1$.
Let $H_2$ be a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G_2$.
Then $H_1 \times H_2... | By definition of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]]:
:$\order {H_1} = p^r$, where $p^r$ is the highest [[Definition:Integer Power|power]] of $p$ which is a [[Definition:Divisor of Integer|divisor]] of $\order {G_1}$.
:$\order {H_2} = p^s$, where $p^s$ is the highest [[Definition:Integer Power|power]] of... | Direct Product of Sylow p-Subgroups is Sylow p-Subgroup | https://proofwiki.org/wiki/Direct_Product_of_Sylow_p-Subgroups_is_Sylow_p-Subgroup | https://proofwiki.org/wiki/Direct_Product_of_Sylow_p-Subgroups_is_Sylow_p-Subgroup | [
"Group Direct Products",
"Sylow p-Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup"
] | [
"Definition:Sylow p-Subgroup",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Sylow p-Subgroup"
] |
proofwiki-15365 | Direct Product of Unique Sylow p-Subgroups is Unique Sylow p-Subgroup | Let $G_1$ and $G_2$ be groups.
Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2$ respectively.
Let $G_1$ be such that $H_1$ is the unique Sylow $p$-subgroup of $G_1$.
Let $G_2$ be such that $H_2$ is the unique Sylow $p$-subgroup of $G_2$.
Then $H_1 \times H_2$ is the unique Sylow $p$-subgroup of $G_1 \times G_2$. | From Direct Product of Sylow p-Subgroups is Sylow p-Subgroup, $H_1 \times H_2$ is a Sylow $p$-subgroup of $G_1 \times G_2$.
By Sylow $p$-Subgroup is Unique iff Normal, each of $H_1$ and $H_2$ are normal in $G_1$ and $G_2$ respectively.
By Direct Product of Normal Subgroups is Normal, $H_1 \times H_2$ is normal in $G_1 ... | Let $G_1$ and $G_2$ be [[Definition:Group|groups]].
Let $H_1$ and $H_2$ be [[Definition:Subgroup|subgroups]] of $G_1$ and $G_2$ respectively.
Let $G_1$ be such that $H_1$ is the [[Definition:Unique|unique]] [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G_1$.
Let $G_2$ be such that $H_2$ is the [[Definition... | From [[Direct Product of Sylow p-Subgroups is Sylow p-Subgroup]], $H_1 \times H_2$ is a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G_1 \times G_2$.
By [[Sylow p-Subgroup is Unique iff Normal|Sylow $p$-Subgroup is Unique iff Normal]], each of $H_1$ and $H_2$ are [[Definition:Normal Subgroup|normal]] in $G_1... | Direct Product of Unique Sylow p-Subgroups is Unique Sylow p-Subgroup | https://proofwiki.org/wiki/Direct_Product_of_Unique_Sylow_p-Subgroups_is_Unique_Sylow_p-Subgroup | https://proofwiki.org/wiki/Direct_Product_of_Unique_Sylow_p-Subgroups_is_Unique_Sylow_p-Subgroup | [
"Group Direct Products",
"Sylow p-Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Unique",
"Definition:Sylow p-Subgroup"
] | [
"Direct Product of Sylow p-Subgroups is Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Normal Subgroup",
"Direct Product of Normal Subgroups is Normal",
"Definition:Normal Subgroup",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Unique",
... |
proofwiki-15366 | Intersection of Sylow p-Subgroup with Subgroup not necessarily Sylow p-Subgroup | Let $G$ be a group.
Let $P$ be a Sylow $p$-subgroup of $G$.
Let $H$ be a subgroup of $G$.
Then $P \cap H$ is not necessarily a Sylow $p$-subgroup of $H$. | We note that from Intersection of Subgroups is Subgroup that $P \cap H$ is a subgroup of $G$ and also of $H$.
Let $G$ be the dihedral group $D_3$, given by its group presentation:
{{:Group Presentation of Dihedral Group D3}}
By definition of Sylow $p$-subgroup, $\gen a$ is a Sylow $3$-subgroup of $G$.
However, $\gen b$... | Let $G$ be a [[Definition:Group|group]].
Let $P$ be a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G$.
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Then $P \cap H$ is not necessarily a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $H$. | We note that from [[Intersection of Subgroups is Subgroup]] that $P \cap H$ is a [[Definition:Subgroup|subgroup]] of $G$ and also of $H$.
Let $G$ be the [[Definition:Dihedral Group D3|dihedral group $D_3$]], given by its [[Group Presentation of Dihedral Group D3|group presentation]]:
{{:Group Presentation of Dihedral... | Intersection of Sylow p-Subgroup with Subgroup not necessarily Sylow p-Subgroup | https://proofwiki.org/wiki/Intersection_of_Sylow_p-Subgroup_with_Subgroup_not_necessarily_Sylow_p-Subgroup | https://proofwiki.org/wiki/Intersection_of_Sylow_p-Subgroup_with_Subgroup_not_necessarily_Sylow_p-Subgroup | [
"Sylow p-Subgroups",
"Set Intersection"
] | [
"Definition:Group",
"Definition:Sylow p-Subgroup",
"Definition:Subgroup",
"Definition:Sylow p-Subgroup"
] | [
"Intersection of Subgroups is Subgroup",
"Definition:Subgroup",
"Definition:Dihedral Group D3",
"Dihedral Group D3/Group Presentation",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup"
] |
proofwiki-15367 | Sylow p-Subgroups of Group of Order 2p | Let $p$ be an odd prime.
Let $G$ be a group of order $2 p$.
Then $G$ has exactly one Sylow $p$-subgroup.
This Sylow $p$-subgroup is normal. | Let $n_p$ denote the number of Sylow $p$-subgroups of $G$.
From the Fourth Sylow Theorem:
:$n_p \equiv 1 \pmod p$
From the Fifth Sylow Theorem:
:$n_p \divides 2 p$
that is:
:$n_p \in \set {1, 2, p, 2 p}$
But $p$ and $2 p$ are congruent to $0$ modulo $p$
So:
:$n_p \notin \set {p, 2 p}$
Also we have that $p > 2$.
Hence:
... | Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $2 p$.
Then $G$ has [[Definition:Unique|exactly one]] [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]].
This [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] is [[Definition:Normal S... | Let $n_p$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] of $G$.
From the [[Fourth Sylow Theorem]]:
:$n_p \equiv 1 \pmod p$
From the [[Fifth Sylow Theorem]]:
:$n_p \divides 2 p$
that is:
:$n_p \in \set {1, 2, p, 2 p}$
But $p$ and $2 p$ are [[Definition:Congruence Modulo Integer|congruent to... | Sylow p-Subgroups of Group of Order 2p/Proof 1 | https://proofwiki.org/wiki/Sylow_p-Subgroups_of_Group_of_Order_2p | https://proofwiki.org/wiki/Sylow_p-Subgroups_of_Group_of_Order_2p/Proof_1 | [
"Sylow p-Subgroups",
"Groups of Order 2 p",
"Sylow p-Subgroups of Group of Order 2p"
] | [
"Definition:Odd Prime",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Congruence (Number Theory)/Integers",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup"
] |
proofwiki-15368 | Sylow p-Subgroups of Group of Order 2p | Let $p$ be an odd prime.
Let $G$ be a group of order $2 p$.
Then $G$ has exactly one Sylow $p$-subgroup.
This Sylow $p$-subgroup is normal. | Let $n_p$ denote the number of Sylow $p$-subgroups of $G$.
From the First Sylow Theorem, there exists at least $1$ Sylow $p$-subgroup of $G$.
Let $P$ be such a Sylow $p$-subgroup of $G$.
The index of $P$ is $2$.
From Subgroup of Index 2 is Normal, $P$ is normal in $G$.
From Sylow $p$-Subgroup is Unique iff Normal it fo... | Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $2 p$.
Then $G$ has [[Definition:Unique|exactly one]] [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]].
This [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] is [[Definition:Normal S... | Let $n_p$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] of $G$.
From the [[First Sylow Theorem]], there exists at least $1$ [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G$.
Let $P$ be such a [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G$.
The [[Definition:Index of Subg... | Sylow p-Subgroups of Group of Order 2p/Proof 2 | https://proofwiki.org/wiki/Sylow_p-Subgroups_of_Group_of_Order_2p | https://proofwiki.org/wiki/Sylow_p-Subgroups_of_Group_of_Order_2p/Proof_2 | [
"Sylow p-Subgroups",
"Groups of Order 2 p",
"Sylow p-Subgroups of Group of Order 2p"
] | [
"Definition:Odd Prime",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Sylow p-Subgroup",
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Index of Subgroup",
"Subgroup of Index 2 is Normal",
"Definition:Normal Subgroup",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Sylow p-Subgroup"
] |
proofwiki-15369 | Sylow p-Subgroups of Group of Order 2p | Let $p$ be an odd prime.
Let $G$ be a group of order $2 p$.
Then $G$ has exactly one Sylow $p$-subgroup.
This Sylow $p$-subgroup is normal. | This is a specific instance of Group of Order $p q$ has Normal Sylow $p$-Subgroup, where $q = 2$.
{{qed}} | Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $2 p$.
Then $G$ has [[Definition:Unique|exactly one]] [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]].
This [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] is [[Definition:Normal S... | This is a specific instance of [[Group of Order p q has Normal Sylow p-Subgroup|Group of Order $p q$ has Normal Sylow $p$-Subgroup]], where $q = 2$.
{{qed}} | Sylow p-Subgroups of Group of Order 2p/Proof 3 | https://proofwiki.org/wiki/Sylow_p-Subgroups_of_Group_of_Order_2p | https://proofwiki.org/wiki/Sylow_p-Subgroups_of_Group_of_Order_2p/Proof_3 | [
"Sylow p-Subgroups",
"Groups of Order 2 p",
"Sylow p-Subgroups of Group of Order 2p"
] | [
"Definition:Odd Prime",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup"
] | [
"Group of Order p q has Normal Sylow p-Subgroup"
] |
proofwiki-15370 | Groups of Order 2p | Let $p$ be a prime number.
Let $G$ be a group.
Let the order of $G$ be $2 p$.
Then $G$ is either:
:the cyclic group $C_{2 p}$
or:
:the dihedral group $D_p$. | When $p = 2$, the result follows from Groups of Order 4.
Let $p$ be an odd prime.
From Sylow p-Subgroups of Group of Order 2p, $G$ has exactly $1$ normal subgroup $P$ of order $p$.
$p$ is prime number.
So from Prime Group is Cyclic, $P$ is a cyclic group.
Let $P = \gen x$ for some $x \in G$.
By the First Sylow Theorem ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Group|group]].
Let the [[Definition:Order of Group|order]] of $G$ be $2 p$.
Then $G$ is either:
:the [[Definition:Cyclic Group|cyclic group $C_{2 p}$]]
or:
:the [[Definition:Dihedral Group|dihedral group $D_p$]]. | When $p = 2$, the result follows from [[Groups of Order 4]].
Let $p$ be an [[Definition:Odd Prime|odd prime]].
From [[Sylow p-Subgroups of Group of Order 2p]], $G$ has [[Definition:Unique|exactly $1$]] [[Definition:Normal Subgroup|normal subgroup]] $P$ of [[Definition:Order of Group|order $p$]].
$p$ is [[Definition... | Groups of Order 2p | https://proofwiki.org/wiki/Groups_of_Order_2p | https://proofwiki.org/wiki/Groups_of_Order_2p | [
"Groups of Order 2 p",
"Finite Cyclic Groups",
"Dihedral Groups"
] | [
"Definition:Prime Number",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Dihedral Group"
] | [
"Groups of Order 4",
"Definition:Odd Prime",
"Sylow p-Subgroups of Group of Order 2p",
"Definition:Unique",
"Definition:Normal Subgroup",
"Definition:Order of Structure",
"Definition:Prime Number",
"Prime Group is Cyclic",
"Definition:Cyclic Group",
"First Sylow Theorem",
"Definition:Subgroup",
... |
proofwiki-15371 | Group of Order p q has Normal Sylow p-Subgroup | Let $p$ and $q$ be prime numbers such that $p > q$.
Let $G$ be a group of order $p q$.
Then $G$ has exactly one Sylow $p$-subgroup.
This Sylow $p$-subgroup is normal. | Let $n_p$ denote the number of Sylow $p$-subgroups in $G$.
From the Fourth Sylow Theorem:
:$n_p \equiv 1 \pmod p$
From the Fifth Sylow Theorem:
:$n_p \divides p q$
where $\divides$ denotes divisibility.
The divisors of $p q$ are $1$, $p$, $q$ and $p q$.
Of these:
:$p$ and $p q$ are $\equiv 0 \pmod p$
and as $p > q$:
:$... | Let $p$ and $q$ be [[Definition:Prime Number|prime numbers]] such that $p > q$.
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $p q$.
Then $G$ has [[Definition:Unique|exactly one]] [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]].
This [[Definition:Sylow p-Subgroup|Sylow $p$-subgro... | Let $n_p$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] in $G$.
From the [[Fourth Sylow Theorem]]:
:$n_p \equiv 1 \pmod p$
From the [[Fifth Sylow Theorem]]:
:$n_p \divides p q$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
The [[Definition:Divisor of Integer|di... | Group of Order p q has Normal Sylow p-Subgroup | https://proofwiki.org/wiki/Group_of_Order_p_q_has_Normal_Sylow_p-Subgroup | https://proofwiki.org/wiki/Group_of_Order_p_q_has_Normal_Sylow_p-Subgroup | [
"Sylow p-Subgroups",
"Examples of Normal Subgroups",
"Groups of Order p q"
] | [
"Definition:Prime Number",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Normal Subgroup"
] |
proofwiki-15372 | Characterisation of Non-Archimedean Division Ring Norms | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Then $\norm {\,\cdot\,}$ is non-Archimedean {{iff}}:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$
where:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dotsb + 1_R}_{\text {$n$ times} }$ | === Necessary Condition ===
{{:Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition}}{{qed|lemma}} | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Then $\norm {\,\cdot\,}$ is [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] {{iff}}:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$
where:
:$n \cd... | === [[Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition|Necessary Condition]] ===
{{:Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition}}{{qed|lemma}} | Characterisation of Non-Archimedean Division Ring Norms | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms | [
"Normed Division Rings",
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Normed Division Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition"
] |
proofwiki-15373 | Groups of Order 21 | There exist exactly $2$ groups of order $21$, up to isomorphism:
:$(1): \quad C_{21}$, the cyclic group of order $21$
:$(2): \quad$ the group whose group presentation is:
:::$\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$ | Let $G$ be of order $21$.
From Group of Order $p q$ has Normal Sylow $p$-Subgroup, $G$ has exactly one Sylow $7$-subgroup, which is normal.
Let this Sylow $7$-subgroup of $G$ be denoted $P = \gen {x: x^7 = 1}$.
From the First Sylow Theorem, $G$ also has at least one Sylow $3$-subgroup.
Thus there exists $y \in G$ of or... | There exist exactly $2$ [[Definition:Group|groups]] of [[Definition:Order of Group|order]] $21$, up to [[Definition:Group Isomorphism|isomorphism]]:
:$(1): \quad C_{21}$, the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order]] $21$
:$(2): \quad$ the [[Definition:Group|group]] whose [[Defin... | Let $G$ be of [[Definition:Order of Group|order]] $21$.
From [[Group of Order p q has Normal Sylow p-Subgroup|Group of Order $p q$ has Normal Sylow $p$-Subgroup]], $G$ has [[Definition:Unique|exactly one]] [[Definition:Sylow p-Subgroup|Sylow $7$-subgroup]], which is [[Definition:Normal Subgroup|normal]].
Let this [[D... | Groups of Order 21 | https://proofwiki.org/wiki/Groups_of_Order_21 | https://proofwiki.org/wiki/Groups_of_Order_21 | [
"Groups of Order 21"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Group",
"Definition:Group Presentation"
] | [
"Definition:Order of Structure",
"Group of Order p q has Normal Sylow p-Subgroup",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup",
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Order of Group Element",
"Definition... |
proofwiki-15374 | Groups of Order 21/Matrix Representation of Non-Abelian Instance | Let $G$ be the group of order $21$ whose group presentation is:
:$\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$
Then $G$ can be instantiated by the following pair of matrices over $\Z_7$:
:$X = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \qquad Y = \begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}$ | We calculate the powers of $X$ and $Y$ in turn:
{{begin-eqn}}
{{eqn | l = X^2
| r = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
| c =
}}
{{eqn | r = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = X^3
| r... | Let $G$ be the [[Definition:Group|group]] of [[Definition:Order of Group|order]] $21$ whose [[Definition:Group Presentation|group presentation]] is:
:$\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$
Then $G$ can be instantiated by the following pair of [[Definition:Square Matrix|matrices]] over $\Z_7$:
:$X = \begin{pm... | We calculate the powers of $X$ and $Y$ in turn:
{{begin-eqn}}
{{eqn | l = X^2
| r = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
| c =
}}
{{eqn | r = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = X^3
|... | Groups of Order 21/Matrix Representation of Non-Abelian Instance | https://proofwiki.org/wiki/Groups_of_Order_21/Matrix_Representation_of_Non-Abelian_Instance | https://proofwiki.org/wiki/Groups_of_Order_21/Matrix_Representation_of_Non-Abelian_Instance | [
"Groups of Order 21"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Group Presentation",
"Definition:Matrix/Square Matrix"
] | [
"Definition:Unit Matrix"
] |
proofwiki-15375 | Normal Sylow p-Subgroups in Group of Order 12 | Let $G$ be of order $12$.
Then $G$ has either:
:a normal Sylow $2$-subgroup
or:
:a normal Sylow $3$-subgroup. | Note that a Sylow $2$-subgroup of $G$ is of order $4$.
From Sylow $3$-Subgroups in Group of Order 12, there are either $1$ or $4$ Sylow $3$-subgroups.
Suppose there is exactly $1$ Sylow $3$-subgroup $P$.
Then from Sylow $p$-Subgroup is Unique iff Normal, $P$ is normal.
{{qed|lemma}}
Suppose there are $4$ Sylow $3$-subg... | Let $G$ be of [[Definition:Order of Group|order]] $12$.
Then $G$ has either:
:a [[Definition:Normal Subgroup|normal]] [[Definition:Sylow p-Subgroup|Sylow $2$-subgroup]]
or:
:a [[Definition:Normal Subgroup|normal]] [[Definition:Sylow p-Subgroup|Sylow $3$-subgroup]]. | Note that a [[Definition:Sylow p-Subgroup|Sylow $2$-subgroup]] of $G$ is of [[Definition:Order of Group|order $4$]].
From [[Sylow Theorems/Examples/Sylow 3-Subgroups in Group of Order 12|Sylow $3$-Subgroups in Group of Order 12]], there are either $1$ or $4$ [[Definition:Sylow p-Subgroup|Sylow $3$-subgroups]].
Suppo... | Normal Sylow p-Subgroups in Group of Order 12 | https://proofwiki.org/wiki/Normal_Sylow_p-Subgroups_in_Group_of_Order_12 | https://proofwiki.org/wiki/Normal_Sylow_p-Subgroups_in_Group_of_Order_12 | [
"Groups of Order 12",
"Sylow p-Subgroups"
] | [
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup"
] | [
"Definition:Sylow p-Subgroup",
"Definition:Order of Structure",
"Sylow Theorems/Examples/Sylow 3-Subgroups in Group of Order 12",
"Definition:Sylow p-Subgroup",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Normal Subgroup",
"Definition:Sylo... |
proofwiki-15376 | Group of Order p^2 q has Normal Sylow p-Subgroup | Let $p$ and $q$ be prime numbers such that $p \ne q$.
Let $G$ be a group of order $p^2 q$.
Then $G$ has a normal Sylow $p$-subgroup. | Let $n_p$ denote the number of Sylow $p$-subgroups in $G$.
From the Fourth Sylow Theorem:
:$n_p \equiv 1 \pmod p$
From the Fifth Sylow Theorem:
:$n_p \divides p^2 q$
where $\divides$ denotes divisibility.
Thus $n_p \in \set {1, q}$.
Suppose $p > q$.
Then:
:$q \not \equiv 1 \pmod p$
and so $n_p \ne q$.
Hence $n_p = 1$.
... | Let $p$ and $q$ be [[Definition:Prime Number|prime numbers]] such that $p \ne q$.
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $p^2 q$.
Then $G$ has a [[Definition:Normal Subgroup|normal]] [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]]. | Let $n_p$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $p$-subgroups]] in $G$.
From the [[Fourth Sylow Theorem]]:
:$n_p \equiv 1 \pmod p$
From the [[Fifth Sylow Theorem]]:
:$n_p \divides p^2 q$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Thus $n_p \in \set {1, q}$.
Suppos... | Group of Order p^2 q has Normal Sylow p-Subgroup | https://proofwiki.org/wiki/Group_of_Order_p^2_q_has_Normal_Sylow_p-Subgroup | https://proofwiki.org/wiki/Group_of_Order_p^2_q_has_Normal_Sylow_p-Subgroup | [
"Sylow p-Subgroups",
"Examples of Normal Subgroups",
"Groups of Order p^2 q"
] | [
"Definition:Prime Number",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup"
] | [
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Sylow p-Subgroup",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Definition:Integral Mu... |
proofwiki-15377 | Group of Order 30 has Normal Cyclic Subgroup of Order 15 | Let $G$ be of order $30$.
Then $G$ has a normal subgroup of order $15$ which is cyclic. | By Group of Order 15 is Cyclic Group, any subgroup of $G$ of order $15$ is cyclic.
It remains to be proved that a subgroup of $G$ of order $15$ exists, and that it is normal.
Let $n_3$ denote the number of Sylow $3$-subgroups of $G$.
From the Fourth Sylow Theorem:
:$n_3 \equiv 1 \pmod 3$
and from the Fifth Sylow Theore... | Let $G$ be of [[Definition:Order of Group|order]] $30$.
Then $G$ has a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $15$ which is [[Definition:Cyclic Group|cyclic]]. | By [[Group of Order 15 is Cyclic Group]], any [[Definition:Subgroup|subgroup]] of $G$ of [[Definition:Order of Group|order]] $15$ is [[Definition:Cyclic Group|cyclic]].
It remains to be proved that a [[Definition:Subgroup|subgroup]] of $G$ of [[Definition:Order of Group|order]] $15$ exists, and that it is [[Definition... | Group of Order 30 has Normal Cyclic Subgroup of Order 15 | https://proofwiki.org/wiki/Group_of_Order_30_has_Normal_Cyclic_Subgroup_of_Order_15 | https://proofwiki.org/wiki/Group_of_Order_30_has_Normal_Cyclic_Subgroup_of_Order_15 | [
"Groups of Order 30"
] | [
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Group of Order 15 is Cyclic Group",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:... |
proofwiki-15378 | Groups of Order 30/Lemma | Let $G$ be a group of order $30$.
Then $G$ is one of the following:
:The cyclic group $C_{30}$
:The dihedral group $D_{15}$
:Isomorphic to one of:
::$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$
::$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$ | By Group of Order 30 has Normal Cyclic Subgroup of Order 15, $G$ has a normal subgroup of order $15$ which is cyclic.
Let this normal cyclic order $15$ subgroup be denoted $N$:
:$N = \gen x$
Let $y$ be the generator for any Sylow $2$-subgroup of $G$.
Then:
{{begin-eqn}}
{{eqn | l = y x y^{-1}
| o = \in
| r ... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $30$.
Then $G$ is one of the following:
:The [[Definition:Cyclic Group|cyclic group]] $C_{30}$
:The [[Definition:Dihedral Group|dihedral group]] $D_{15}$
:[[Definition:Group Isomorphism|Isomorphic]] to one of:
::$\gen {x, y: x^{15} = e ... | By [[Group of Order 30 has Normal Cyclic Subgroup of Order 15]], $G$ has a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $15$ which is [[Definition:Cyclic Group|cyclic]].
Let this [[Definition:Normal Subgroup|normal]] [[Definition:Cyclic Group|cyclic]] [[Definition:Order of Grou... | Groups of Order 30/Lemma | https://proofwiki.org/wiki/Groups_of_Order_30/Lemma | https://proofwiki.org/wiki/Groups_of_Order_30/Lemma | [
"Groups of Order 30"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Dihedral Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Group of Order 30 has Normal Cyclic Subgroup of Order 15",
"Definition:Normal Subgroup",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Normal Subgroup",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Subgroup",
"Definition:Generator of Subgroup",
... |
proofwiki-15379 | Normal Subgroup of Group of Order 24 | Let $G$ be a group of order $24$.
Then $G$ has either:
:a normal subgroup of order $8$
or:
:a normal subgroup of order $4$. | We note that:
:$24 = 3 \times 2^3$
Hence a Sylow $2$-subgroup of $G$ is of order $8$.
Let $n_2$ denote the number of Sylow $2$-subgroups of $G$.
By the Fourth Sylow Theorem:
:$n_2 \equiv 1 \pmod 2$ (that is, $n_2$ is odd
and from the Fifth Sylow Theorem:
:$n_2 \divides 24$
where $\divides$ denotes divisibility.
It foll... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $24$.
Then $G$ has either:
:a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $8$
or:
:a [[Definition:Normal Subgroup|normal subgroup]] of [[Definition:Order of Group|order]] $4$. | We note that:
:$24 = 3 \times 2^3$
Hence a [[Definition:Sylow p-Subgroup|Sylow $2$-subgroup]] of $G$ is of [[Definition:Order of Group|order]] $8$.
Let $n_2$ denote the number of [[Definition:Sylow p-Subgroup|Sylow $2$-subgroups]] of $G$.
By the [[Fourth Sylow Theorem]]:
:$n_2 \equiv 1 \pmod 2$ (that is, $n_2$ is [... | Normal Subgroup of Group of Order 24 | https://proofwiki.org/wiki/Normal_Subgroup_of_Group_of_Order_24 | https://proofwiki.org/wiki/Normal_Subgroup_of_Group_of_Order_24 | [
"Groups of Order 24"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Order of Structure"
] | [
"Definition:Sylow p-Subgroup",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Definition:Odd Integer",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"Definition:Unique",
"Definition:Sylow p-Subgroup",
"Sylow p-Subgroup is Unique iff Normal... |
proofwiki-15380 | Group of Order 35 is Cyclic Group | Let $G$ be a group whose order is $35$.
Then $G$ is cyclic. | We have that $35 = 5 \times 7$.
Then we have that $5$ and $7$ are primes such that $5 < 7$ and $5$ does not divide $7 - 1$.
Thus Group of Order $p q$ is Cyclic can be applied.
{{Qed}} | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $35$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | We have that $35 = 5 \times 7$.
Then we have that $5$ and $7$ are [[Definition:Prime Number|primes]] such that $5 < 7$ and $5$ does not [[Definition:Divisor of Integer|divide]] $7 - 1$.
Thus [[Group of Order p q is Cyclic|Group of Order $p q$ is Cyclic]] can be applied.
{{Qed}} | Group of Order 35 is Cyclic Group/Proof 1 | https://proofwiki.org/wiki/Group_of_Order_35_is_Cyclic_Group | https://proofwiki.org/wiki/Group_of_Order_35_is_Cyclic_Group/Proof_1 | [
"Groups of Order 35",
"Finite Cyclic Groups",
"Group of Order 35 is Cyclic Group"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Group of Order p q is Cyclic"
] |
proofwiki-15381 | Group of Order 35 is Cyclic Group | Let $G$ be a group whose order is $35$.
Then $G$ is cyclic. | Let $G$ be of order $35$.
We have that $35 = 5 \times 7$ where both $5$ and $7$ are prime.
Hence from the First Sylow Theorem, $G$ has:
:at least one Sylow $5$-subgroup
and:
:at least one Sylow $7$-subgroup
Let $n_5$ denote the number of Sylow $5$-subgroups of $G$.
Let $n_7$ denote the number of Sylow $7$-subgroups of ... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $35$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | Let $G$ be of [[Definition:Order of Group|order $35$]].
We have that $35 = 5 \times 7$ where both $5$ and $7$ are [[Definition:Prime Number|prime]].
Hence from the [[First Sylow Theorem]], $G$ has:
:at least one [[Definition:Sylow p-Subgroup|Sylow $5$-subgroup]]
and:
:at least one [[Definition:Sylow p-Subgroup|Sylow ... | Group of Order 35 is Cyclic Group/Proof 2 | https://proofwiki.org/wiki/Group_of_Order_35_is_Cyclic_Group | https://proofwiki.org/wiki/Group_of_Order_35_is_Cyclic_Group/Proof_2 | [
"Groups of Order 35",
"Finite Cyclic Groups",
"Group of Order 35 is Cyclic Group"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Definition:Order of Structure",
"Definition:Prime Number",
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Fourth Sylow Theorem",
"Fifth Sylow Theorem",
"Definition:Divisor (Algebra)/Integer",
"D... |
proofwiki-15382 | Group of Order 105 has Normal Sylow 5-Subgroup or Normal Sylow 7-Subgroup | Let $G$ be a group of order $105$.
Then $G$ has either:
:exactly one normal Sylow $5$-subgroup
or:
:exactly one normal Sylow $7$-subgroup. | Let $G$ be a group of order $105$ whose identity is $e$.
We have that:
:$105 = 3 \times 5 \times 7$
From the First Sylow Theorem, $G$ has at least one Sylow $3$-subgroup, Sylow $5$-subgroup and Sylow $7$-subgroup.
Let:
:$n_5$ denote the number of Sylow $5$-subgroups of $G$
:$n_7$ denote the number of Sylow $7$-subgroup... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $105$.
Then $G$ has either:
:[[Definition:Unique|exactly one]] [[Definition:Normal Subgroup|normal]] [[Definition:Sylow p-Subgroup|Sylow $5$-subgroup]]
or:
:[[Definition:Unique|exactly one]] [[Definition:Normal Subgroup|normal]] [[Definitio... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $105$ whose [[Definition:Identity Element|identity]] is $e$.
We have that:
:$105 = 3 \times 5 \times 7$
From the [[First Sylow Theorem]], $G$ has at least one [[Definition:Sylow p-Subgroup|Sylow $3$-subgroup]], [[Definition:Sylow p-Subgrou... | Group of Order 105 has Normal Sylow 5-Subgroup or Normal Sylow 7-Subgroup | https://proofwiki.org/wiki/Group_of_Order_105_has_Normal_Sylow_5-Subgroup_or_Normal_Sylow_7-Subgroup | https://proofwiki.org/wiki/Group_of_Order_105_has_Normal_Sylow_5-Subgroup_or_Normal_Sylow_7-Subgroup | [
"Groups of Order 105",
"Sylow p-Subgroups",
"Examples of Normal Subgroups"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Unique",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Unique",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Defin... |
proofwiki-15383 | Group of Order 105 has Normal Cyclic Subgroup of Index 3 | Let $G$ be a group of order $105$.
Then $G$ has a normal cyclic subgroup $N$ such that:
:$\index G N = 3$
where $\index G N$ denotes the index of $N$ in $G$. | Let $G$ be a group of order $105$ whose identity is $e$.
From Group of Order 105 has Normal Sylow 5-Subgroup or Normal Sylow 7-Subgroup, $G$ has either:
:exactly one normal Sylow $5$-subgroup
or:
:exactly one normal Sylow $7$-subgroup.
Suppose $G$ has exactly one normal Sylow $5$-subgroup, which we will denote $P$.
The... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $105$.
Then $G$ has a [[Definition:Normal Subgroup|normal]] [[Definition:Cyclic Group|cyclic]] [[Definition:Subgroup|subgroup]] $N$ such that:
:$\index G N = 3$
where $\index G N$ denotes the [[Definition:Index of Subgroup|index]] of $N$ in... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $105$ whose [[Definition:Identity Element|identity]] is $e$.
From [[Group of Order 105 has Normal Sylow 5-Subgroup or Normal Sylow 7-Subgroup]], $G$ has either:
:[[Definition:Unique|exactly one]] [[Definition:Normal Subgroup|normal]] [[Def... | Group of Order 105 has Normal Cyclic Subgroup of Index 3 | https://proofwiki.org/wiki/Group_of_Order_105_has_Normal_Cyclic_Subgroup_of_Index_3 | https://proofwiki.org/wiki/Group_of_Order_105_has_Normal_Cyclic_Subgroup_of_Index_3 | [
"Groups of Order 105",
"Examples of Normal Subgroups"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Normal Subgroup",
"Definition:Cyclic Group",
"Definition:Subgroup",
"Definition:Index of Subgroup"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Group of Order 105 has Normal Sylow 5-Subgroup or Normal Sylow 7-Subgroup",
"Definition:Unique",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Unique",
"Defin... |
proofwiki-15384 | Diagonal Relation is Reflexive | The diagonal relation $\Delta_S$ on a set $S$ is a reflexive relation in $S$. | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = x
| r = x
| c = {{Defof|Equals}}
}}
{{eqn | ll= \leadsto
| l = \tuple {x, x}
| o = \in
| r = \Delta_S
| c = {{Defof|Diagonal Relation}}
}}
{{end-eqn}}
So $\Delta_S$ is reflexive. | The [[Definition:Diagonal Relation|diagonal relation]] $\Delta_S$ on a [[Definition:Set|set]] $S$ is a [[Definition:Reflexive Relation|reflexive relation]] in $S$. | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = x
| r = x
| c = {{Defof|Equals}}
}}
{{eqn | ll= \leadsto
| l = \tuple {x, x}
| o = \in
| r = \Delta_S
| c = {{Defof|Diagonal Relation}}
}}
{{end-eqn}}
So $\Delta_S$ is [[Definition:Reflexive Relation|reflexive]]. | Diagonal Relation is Reflexive | https://proofwiki.org/wiki/Diagonal_Relation_is_Reflexive | https://proofwiki.org/wiki/Diagonal_Relation_is_Reflexive | [
"Diagonal Relation is Equivalence",
"Diagonal Relation",
"Examples of Reflexive Relations"
] | [
"Definition:Diagonal Relation",
"Definition:Set",
"Definition:Reflexive Relation"
] | [
"Definition:Reflexive Relation"
] |
proofwiki-15385 | Diagonal Relation is Symmetric | The diagonal relation $\Delta_S$ on a set $S$ is a symmetric relation in $S$. | {{begin-eqn}}
{{eqn | q = \forall x, y \in S
| l = \tuple {x, y}
| o = \in
| r = \Delta_S
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y
| c = {{Defof|Diagonal Relation}}
}}
{{eqn | ll= \leadsto
| l = y
| r = x
| c = Equality is Symmetric
}}
{{eqn | ll= \leads... | The [[Definition:Diagonal Relation|diagonal relation]] $\Delta_S$ on a [[Definition:Set|set]] $S$ is a [[Definition:Symmetric Relation|symmetric relation]] in $S$. | {{begin-eqn}}
{{eqn | q = \forall x, y \in S
| l = \tuple {x, y}
| o = \in
| r = \Delta_S
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y
| c = {{Defof|Diagonal Relation}}
}}
{{eqn | ll= \leadsto
| l = y
| r = x
| c = [[Equality is Symmetric]]
}}
{{eqn | ll= \l... | Diagonal Relation is Symmetric | https://proofwiki.org/wiki/Diagonal_Relation_is_Symmetric | https://proofwiki.org/wiki/Diagonal_Relation_is_Symmetric | [
"Diagonal Relation is Equivalence",
"Diagonal Relation",
"Examples of Symmetric Relations"
] | [
"Definition:Diagonal Relation",
"Definition:Set",
"Definition:Symmetric Relation"
] | [
"Equality is Symmetric",
"Definition:Symmetric Relation"
] |
proofwiki-15386 | Diagonal Relation is Transitive | The diagonal relation $\Delta_S$ on a set $S$ is a transitive relation in $S$. | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = \tuple {x, y}
| o = \in
| r = \Delta_S \land \tuple {y, z} \in \Delta_S
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y \land y = z
| c = {{Defof|Diagonal Relation}}
}}
{{eqn | ll= \leadsto
| l = x
| r = z
|... | The [[Definition:Diagonal Relation|diagonal relation]] $\Delta_S$ on a [[Definition:Set|set]] $S$ is a [[Definition:Transitive Relation|transitive relation]] in $S$. | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = \tuple {x, y}
| o = \in
| r = \Delta_S \land \tuple {y, z} \in \Delta_S
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y \land y = z
| c = {{Defof|Diagonal Relation}}
}}
{{eqn | ll= \leadsto
| l = x
| r = z
|... | Diagonal Relation is Transitive | https://proofwiki.org/wiki/Diagonal_Relation_is_Transitive | https://proofwiki.org/wiki/Diagonal_Relation_is_Transitive | [
"Diagonal Relation is Equivalence",
"Diagonal Relation",
"Examples of Transitive Relations"
] | [
"Definition:Diagonal Relation",
"Definition:Set",
"Definition:Transitive Relation"
] | [
"Equality is Transitive",
"Definition:Transitive Relation"
] |
proofwiki-15387 | Group of Order 56 has Unique Sylow 2-Subgroup or Unique Sylow 7-Subgroup | Let $G$ be a group of order $56$.
Then $G$ has either:
:exactly one normal Sylow $2$-subgroup
or:
:exactly one normal Sylow $7$-subgroup. | Let $G$ be a group of order $56$ whose identity is $e$.
We have that:
:$56 = 2^3 \times 7$
From the First Sylow Theorem, $G$ has at least one Sylow $2$-subgroup and Sylow $7$-subgroup.
Let:
:$n_2$ denote the number of Sylow $2$-subgroups of $G$
:$n_7$ denote the number of Sylow $7$-subgroups of $G$.
From Sylow p-Subgro... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $56$.
Then $G$ has either:
:[[Definition:Unique|exactly one]] [[Definition:Normal Subgroup|normal]] [[Definition:Sylow p-Subgroup|Sylow $2$-subgroup]]
or:
:[[Definition:Unique|exactly one]] [[Definition:Normal Subgroup|normal]] [[Definition... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $56$ whose [[Definition:Identity Element|identity]] is $e$.
We have that:
:$56 = 2^3 \times 7$
From the [[First Sylow Theorem]], $G$ has at least one [[Definition:Sylow p-Subgroup|Sylow $2$-subgroup]] and [[Definition:Sylow p-Subgroup|Sylo... | Group of Order 56 has Unique Sylow 2-Subgroup or Unique Sylow 7-Subgroup | https://proofwiki.org/wiki/Group_of_Order_56_has_Unique_Sylow_2-Subgroup_or_Unique_Sylow_7-Subgroup | https://proofwiki.org/wiki/Group_of_Order_56_has_Unique_Sylow_2-Subgroup_or_Unique_Sylow_7-Subgroup | [
"Groups of Order 56",
"Sylow p-Subgroups",
"Examples of Normal Subgroups"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Unique",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Unique",
"Definition:Normal Subgroup",
"Definition:Sylow p-Subgroup"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"First Sylow Theorem",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Sylow p-Subgroup is Unique iff Normal"... |
proofwiki-15388 | Subgroup of Direct Product is not necessarily Direct Product of Subgroups | Let $G$ and $H$ be groups.
Let $G \times H$ denote the direct product of $G$ and $H$.
Let $K$ be a subgroup of $G \times H$.
Then it is not necessarily the case that $K$ is of the form:
:$G' \times H'$
where:
:$G'$ is a subgroup of $G$
:$H'$ is a subgroup of $H$. | Let $G = H = C_2$, the cyclic group of order $2$.
Let $G = \gen x$ and $H = \gen y$, so that:
:$G = \set {e_G, x}$
:$H = \set {e_H, y}$
where $e_G$ and $e_H$ are the identity elements of $G$ and $H$ respectively.
Consider the element $\tuple {x, y} \in G \times H$.
We have that:
:$\gen {\tuple {x, y} } =\set {\tuple {e... | Let $G$ and $H$ be [[Definition:Group|groups]].
Let $G \times H$ denote the [[Definition:Group Direct Product|direct product]] of $G$ and $H$.
Let $K$ be a [[Definition:Subgroup|subgroup]] of $G \times H$.
Then it is not necessarily the case that $K$ is of the form:
:$G' \times H'$
where:
:$G'$ is a [[Definition:Su... | Let $G = H = C_2$, the [[Definition:Cyclic Group|cyclic group]] of [[Definition:Order of Group|order $2$]].
Let $G = \gen x$ and $H = \gen y$, so that:
:$G = \set {e_G, x}$
:$H = \set {e_H, y}$
where $e_G$ and $e_H$ are the [[Definition:Identity Element|identity elements]] of $G$ and $H$ respectively.
Consider the... | Subgroup of Direct Product is not necessarily Direct Product of Subgroups | https://proofwiki.org/wiki/Subgroup_of_Direct_Product_is_not_necessarily_Direct_Product_of_Subgroups | https://proofwiki.org/wiki/Subgroup_of_Direct_Product_is_not_necessarily_Direct_Product_of_Subgroups | [
"Group Direct Products"
] | [
"Definition:Group",
"Definition:Group Direct Product",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Subgroup"
] | [
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Group Direct Product",
"Definition:Subgroup",
"Definition:Subgroup"
] |
proofwiki-15389 | Groups of Order 30 | Let $G$ be a group of order $30$.
Then $G$ is one of the following:
:The cyclic group $C_{30}$
:The dihedral group $D_{15}$
:The group direct product $C_5 \times D_3$
:The group direct product $C_3 \times D_5$ | First we introduce a lemma: | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $30$.
Then $G$ is one of the following:
:The [[Definition:Cyclic Group|cyclic group]] $C_{30}$
:The [[Definition:Dihedral Group|dihedral group]] $D_{15}$
:The [[Definition:Group Direct Product|group direct product]] $C_5 \times D_3$
:Th... | First we introduce a [[Definition:Lemma|lemma]]: | Groups of Order 30 | https://proofwiki.org/wiki/Groups_of_Order_30 | https://proofwiki.org/wiki/Groups_of_Order_30 | [
"Groups of Order 30"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Dihedral Group",
"Definition:Group Direct Product",
"Definition:Group Direct Product"
] | [
"Definition:Lemma"
] |
proofwiki-15390 | Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Then:
:$\norm {\,\cdot\,}$ is non-Archimedean $\implies \forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$.
where:
$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$ | Let $\norm {\,\cdot\,}$ be non-Archimedean.
Then by the definition of a non-Archimedean norm, for $n \in \N$:
{{begin-eqn}}
{{eqn | q = \forall n \in \N_{>0}
| l = \norm {n \cdot 1_R}
| r = \norm {1_R + \dots + 1_R}
| c = ($n$ summands)
}}
{{eqn | o = \le
| r = \max \set {\norm {1_R}, \ldots, \n... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Then:
:$\norm {\,\cdot\,}$ is [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] $\implies \forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$.
where:
$n \cd... | Let $\norm {\,\cdot\,}$ be [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]].
Then by the definition of a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]], for $n \in \N$:
{{begin-eqn}}
{{eqn | q = \forall n \in \N_{>0}
| l = \norm {n \cdot 1_R}
| r = \norm {1_R + \do... | Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Necessary_Condition | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Necessary_Condition | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Normed Division Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)"
] |
proofwiki-15391 | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Then:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1 \implies \norm {\,\cdot\,}$ is non-archimedean
where:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \text { times} }$ | Let:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$
Let $x, y \in R$.
Let $y = 0_R$ where $0_R$ is the zero of $R$.
Then $\norm {x + y} = \norm x = \max \set {\norm x, 0} = \max \set {\norm x, \norm y}$
==== Lemma 1 ====
{{:Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1}}{{qed... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Then:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1 \implies \norm {\,\cdot\,}$ is [[Definition:Non-Archimedean Division Ring Norm|non-archimedean]]
where:
:$n \cdot... | Let:
:$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$
Let $x, y \in R$.
Let $y = 0_R$ where $0_R$ is the [[Definition:Ring Zero|zero]] of $R$.
Then $\norm {x + y} = \norm x = \max \set {\norm x, 0} = \max \set {\norm x, \norm y}$
==== [[Characterisation of Non-Archimedean Division Ring Norms/Sufficient Conditi... | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Normed Division Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Ring Zero",
"Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1",
"Binomial Theorem",
"Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2",
"Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3",
... |
proofwiki-15392 | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1 | Let $y \ne 0_R$ where $0_R$ is the zero of $R$.
Then:
:$\norm {x + y} \le \max \set {\norm x, \norm y} \iff \norm {x y^{-1} + 1_R} \le \max \set {\norm {x y^{-1} }, 1}$ | {{begin-eqn}}
{{eqn | l = \norm {x + y}
| o = \le
| r = \max \set {\norm x, \norm y}
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x + y} \norm {y^{-1} }
| o = \le
| r = \max \set {\norm x \norm {y^{-1} }, \norm y \norm {y^{-1} } }
| c = Multiply through by $\norm{y^{-1} }$
}}
{{eqn | ... | Let $y \ne 0_R$ where $0_R$ is the [[Definition:Ring Zero|zero]] of $R$.
Then:
:$\norm {x + y} \le \max \set {\norm x, \norm y} \iff \norm {x y^{-1} + 1_R} \le \max \set {\norm {x y^{-1} }, 1}$ | {{begin-eqn}}
{{eqn | l = \norm {x + y}
| o = \le
| r = \max \set {\norm x, \norm y}
}}
{{eqn | ll= \leadstoandfrom
| l = \norm {x + y} \norm {y^{-1} }
| o = \le
| r = \max \set {\norm x \norm {y^{-1} }, \norm y \norm {y^{-1} } }
| c = Multiply through by $\norm{y^{-1} }$
}}
{{eqn | ... | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition/Lemma_1 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition/Lemma_1 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Ring Zero"
] | [
"Definition:Division Ring",
"Properties of Norm on Division Ring/Norm of Unity"
] |
proofwiki-15393 | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2 | Then for all $i$, $0 \le i \le n$:
:$\norm x^i \le \max \set {\norm x^n , 1}$ | If $\norm x > 1$ then for all $i$, $0 \le i \le n$:
:$\norm x^i \le \norm x^n \le \max \set {\norm x^n, 1}$
If $\norm x \le 1$ then for all $i$, $0 \le i \le n$:
:$\norm x^i \le 1 \le \max \set {\norm x^n, 1}$
In either case for all $i$, $0 \le i \le n$:
:$\norm x^i \le \max \set {\norm x^n , 1}$
{{qed}} | Then for all $i$, $0 \le i \le n$:
:$\norm x^i \le \max \set {\norm x^n , 1}$ | If $\norm x > 1$ then for all $i$, $0 \le i \le n$:
:$\norm x^i \le \norm x^n \le \max \set {\norm x^n, 1}$
If $\norm x \le 1$ then for all $i$, $0 \le i \le n$:
:$\norm x^i \le 1 \le \max \set {\norm x^n, 1}$
In either case for all $i$, $0 \le i \le n$:
:$\norm x^i \le \max \set {\norm x^n , 1}$
{{qed}} | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition/Lemma_2 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition/Lemma_2 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [] | [] |
proofwiki-15394 | Group Epimorphism preserves Central Subgroups | Let $G$ and $H$ be groups.
Let $\theta: G \to H$ be an epimorphism.
Let $Z \le G$ be a central subgroup of $G$.
Then $\theta \sqbrk Z$ is a central subgroup of $H$. | By definition of central subgroup:
:$Z \subseteq \map Z G$
where $\map Z G$ denotes the center of $G$.
From Image under Epimorphism of Center is Subset of Center:
:$\theta \sqbrk {\map Z G} \subseteq \map Z H$
From Image of Subset under Mapping is Subset of Image it follows that:
:$\theta \sqbrk Z \subseteq \map Z H$
T... | Let $G$ and $H$ be [[Definition:Group|groups]].
Let $\theta: G \to H$ be an [[Definition:Group Epimorphism|epimorphism]].
Let $Z \le G$ be a [[Definition:Central Subgroup|central subgroup]] of $G$.
Then $\theta \sqbrk Z$ is a [[Definition:Central Subgroup|central subgroup]] of $H$. | By definition of [[Definition:Central Subgroup|central subgroup]]:
:$Z \subseteq \map Z G$
where $\map Z G$ denotes the [[Definition:Center of Group|center]] of $G$.
From [[Image under Epimorphism of Center is Subset of Center]]:
:$\theta \sqbrk {\map Z G} \subseteq \map Z H$
From [[Image of Subset under Mapping is S... | Group Epimorphism preserves Central Subgroups | https://proofwiki.org/wiki/Group_Epimorphism_preserves_Central_Subgroups | https://proofwiki.org/wiki/Group_Epimorphism_preserves_Central_Subgroups | [
"Central Subgroups",
"Group Epimorphisms"
] | [
"Definition:Group",
"Definition:Group Epimorphism",
"Definition:Central Subgroup",
"Definition:Central Subgroup"
] | [
"Definition:Central Subgroup",
"Definition:Center (Abstract Algebra)/Group",
"Image under Epimorphism of Center is Subset of Center",
"Image of Subset under Mapping is Subset of Image",
"Category:Central Subgroups",
"Category:Group Epimorphisms"
] |
proofwiki-15395 | Direct Product of Central Subgroups | Let $G$ and $H$ be groups.
Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively.
Then $Z \times W$ is a central subgroup of $G \times H$. | Let $\tuple {z, w} \in Z \times W$.
Let $\tuple {x, y} \in G \times H$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, y} \tuple {z, w}
| r = \tuple {x z, y w}
| c = {{Defof|Group Direct Product}}
}}
{{eqn | r = \tuple {z x, w y}
| c = {{Defof|Central Subgroup}}
}}
{{eqn | r = \tuple {z, w} \tuple {x, y}
... | Let $G$ and $H$ be [[Definition:Group|groups]].
Let $Z$ and $W$ be [[Definition:Central Subgroup|central subgroups]] of $G$ and $H$ respectively.
Then $Z \times W$ is a [[Definition:Central Subgroup|central subgroup]] of $G \times H$. | Let $\tuple {z, w} \in Z \times W$.
Let $\tuple {x, y} \in G \times H$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, y} \tuple {z, w}
| r = \tuple {x z, y w}
| c = {{Defof|Group Direct Product}}
}}
{{eqn | r = \tuple {z x, w y}
| c = {{Defof|Central Subgroup}}
}}
{{eqn | r = \tuple {z, w} \tuple {x, ... | Direct Product of Central Subgroups | https://proofwiki.org/wiki/Direct_Product_of_Central_Subgroups | https://proofwiki.org/wiki/Direct_Product_of_Central_Subgroups | [
"Central Subgroups",
"Group Direct Products"
] | [
"Definition:Group",
"Definition:Central Subgroup",
"Definition:Central Subgroup"
] | [
"Definition:Commutative/Elements",
"Definition:Element",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Subgroup",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Central Subgroup",
"Category:Central Subgroups",
"Category:Group Direct Products"
] |
proofwiki-15396 | Groups of Order 30/C 5 x D 3 | Let $G$ be a group of order $30$.
Let $G$ have the group presentation:
:$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$
Then $G$ is isomorphic to the group direct product of the cyclic group $C_5$ and the dihedral group $D_3$:
:$G \cong C_5 \times D_3$ | Let $G$ be defined by its group presentation:
:$G = \gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$
Let $z$ denote $x^3$.
Then:
{{begin-eqn}}
{{eqn | l = y z y^{-1}
| r = y x^3 y^{-1}
| c =
}}
{{eqn | r = \paren {y x y^{-1} }^3
| c = Power of Conjugate equals Conjugate of Power
}}
{{eqn | r = \p... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $30$.
Let $G$ have the [[Definition:Group Presentation|group presentation]]:
:$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$
Then $G$ is [[Definition:Group Isomorphism|isomorphic]] to the [[Definition:Group Direct Product|group di... | Let $G$ be defined by its [[Definition:Group Presentation|group presentation]]:
:$G = \gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$
Let $z$ denote $x^3$.
Then:
{{begin-eqn}}
{{eqn | l = y z y^{-1}
| r = y x^3 y^{-1}
| c =
}}
{{eqn | r = \paren {y x y^{-1} }^3
| c = [[Power of Conjugate eq... | Groups of Order 30/C 5 x D 3 | https://proofwiki.org/wiki/Groups_of_Order_30/C_5_x_D_3 | https://proofwiki.org/wiki/Groups_of_Order_30/C_5_x_D_3 | [
"Groups of Order 30"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Group Presentation",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Group Direct Product",
"Definition:Cyclic Group",
"Definition:Dihedral Group"
] | [
"Definition:Group Presentation",
"Power of Conjugate equals Conjugate of Power",
"Definition:Group Presentation",
"Powers of Group Elements/Product of Indices",
"Powers of Group Elements/Sum of Indices",
"Definition:Group Presentation",
"Definition:Group Product/Product Element",
"Definition:Commutati... |
proofwiki-15397 | Groups of Order 30/C 3 x D 5 | Let $G$ be a group of order $30$.
Let $G$ have the group presentation:
:$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$
Then $G$ is isomorphic to the group direct product of the cyclic group $C_3$ and the dihedral group $D_5$:
:$G \cong C_3 \times D_5$ | Let $G$ be defined by its group presentation:
:$G = \gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$
Let $z$ denote $x^5$.
Then:
{{begin-eqn}}
{{eqn | l = y z y^{-1}
| r = y x^5 y^{-1}
| c =
}}
{{eqn | r = \paren {y x y^{-1} }^5
| c = Power of Conjugate equals Conjugate of Power
}}
{{eqn | r = \paren... | Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Group|order]] $30$.
Let $G$ have the [[Definition:Group Presentation|group presentation]]:
:$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$
Then $G$ is [[Definition:Group Isomorphism|isomorphic]] to the [[Definition:Group Direct Product|group direct... | Let $G$ be defined by its [[Definition:Group Presentation|group presentation]]:
:$G = \gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$
Let $z$ denote $x^5$.
Then:
{{begin-eqn}}
{{eqn | l = y z y^{-1}
| r = y x^5 y^{-1}
| c =
}}
{{eqn | r = \paren {y x y^{-1} }^5
| c = [[Power of Conjugate equals... | Groups of Order 30/C 3 x D 5 | https://proofwiki.org/wiki/Groups_of_Order_30/C_3_x_D_5 | https://proofwiki.org/wiki/Groups_of_Order_30/C_3_x_D_5 | [
"Groups of Order 30"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Group Presentation",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Group Direct Product",
"Definition:Cyclic Group",
"Definition:Dihedral Group"
] | [
"Definition:Group Presentation",
"Power of Conjugate equals Conjugate of Power",
"Definition:Group Presentation",
"Powers of Group Elements/Product of Indices",
"Powers of Group Elements/Sum of Indices",
"Definition:Group Presentation",
"Definition:Group Product/Product Element",
"Definition:Commutati... |
proofwiki-15398 | Dihedral Group D6 is Internal Direct Product of C2 with D3 | The dihedral group $D_6$ is an internal direct product of the cyclic group $C_2$ of order $2$ and the dihedral group $D_3$:
:$D_6 = C_2 \times D_3$ | Let $G$ be defined by its group presentation:
:$G = \gen {x, y: x^6 = e = y^2, y x y^{-1} = x^{-1} }$
or:
:$G = \gen {x, y: x^6 = e = y^2, y x y^{-1} = x^5}$
Let $z$ denote $x^3$.
Then:
{{begin-eqn}}
{{eqn | l = y z y^{-1}
| r = y x^3 y^{-1}
| c =
}}
{{eqn | r = \paren {y x y^{-1} }^3
| c = Power of ... | The [[Definition:Dihedral Group D6|dihedral group $D_6$]] is an [[Definition:Internal Group Direct Product|internal direct product]] of the [[Definition:Cyclic Group|cyclic group]] $C_2$ of [[Definition:Order of Group|order $2$]] and the [[Definition:Dihedral Group D3|dihedral group $D_3$]]:
:$D_6 = C_2 \times D_3$ | Let $G$ be defined by its [[Definition:Group Presentation|group presentation]]:
:$G = \gen {x, y: x^6 = e = y^2, y x y^{-1} = x^{-1} }$
or:
:$G = \gen {x, y: x^6 = e = y^2, y x y^{-1} = x^5}$
Let $z$ denote $x^3$.
Then:
{{begin-eqn}}
{{eqn | l = y z y^{-1}
| r = y x^3 y^{-1}
| c =
}}
{{eqn | r = \pare... | Dihedral Group D6 is Internal Direct Product of C2 with D3 | https://proofwiki.org/wiki/Dihedral_Group_D6_is_Internal_Direct_Product_of_C2_with_D3 | https://proofwiki.org/wiki/Dihedral_Group_D6_is_Internal_Direct_Product_of_C2_with_D3 | [
"Dihedral Group D6",
"Examples of Internal Group Direct Products"
] | [
"Definition:Dihedral Group D6",
"Definition:Internal Group Direct Product",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Dihedral Group D3"
] | [
"Definition:Group Presentation",
"Power of Conjugate equals Conjugate of Power",
"Powers of Group Elements/Product of Indices",
"Powers of Group Elements/Sum of Indices",
"Definition:Group Product/Product Element",
"Definition:Commutative/Elements",
"Definition:Power of Element/Group",
"Definition:Com... |
proofwiki-15399 | Sequence of Integers defining Abelian Group | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $C_n$ be a finite abelian group.
Then $C_n$ is of the form:
:$C_{n_1} \times C_{n_2} \times \cdots \times C_{n_r}$
such that:
:$n = \ds \prod_{k \mathop = 1}^r n_k$
:$\forall k \in \set {2, 3, \ldots, r}: n_k \divides n_{k - 1}$
where $\divides$ denotes divisibili... | {{ProofWanted|This is probably just a statement of Fundamental Theorem of Finite Abelian Groups, which needs to be studied to see what it actually means}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $C_n$ be a [[Definition:Finite Group|finite]] [[Definition:Abelian Group|abelian group]].
Then $C_n$ is of the form:
:$C_{n_1} \times C_{n_2} \times \cdots \times C_{n_r}$
such that:
:$n = \ds \prod_{k \mathop = 1}^r ... | {{ProofWanted|This is probably just a statement of [[Fundamental Theorem of Finite Abelian Groups]], which needs to be studied to see what it actually means}} | Sequence of Integers defining Abelian Group | https://proofwiki.org/wiki/Sequence_of_Integers_defining_Abelian_Group | https://proofwiki.org/wiki/Sequence_of_Integers_defining_Abelian_Group | [
"Abelian Groups",
"Sequence of Integers defining Abelian Group"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Finite Group",
"Definition:Abelian Group",
"Definition:Divisor (Algebra)/Integer"
] | [
"Fundamental Theorem of Finite Abelian Groups"
] |
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