hcju/proofwikips huggingface dataset:
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: :$\norm x < 1$  $\sequence {x_n}$ is a null sequence.
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: :$\size x < 1$  $\sequence {x_n}$ is a null sequence.
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 $G$.
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}}
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.
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 sequence in $\norm {\, \cdot \,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_2$
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 $\forall x \in R$: :$\norm x_1 < 1 \iff \norm x_2 < 1$
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: :$\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
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 of $P$. Then $P Q$ is not a subgroup of $G$.
Linearly Independent Set is Contained in some Basis
There exists a basis $B$ for $E$ such that $H \subseteq B$.
Linearly Independent Set is Basis iff of Same Cardinality as Dimension
$H$ is a basis for $E$  it contains exactly $n$ elements.
ISBN-10 is Error-Correcting Code
ISBN-$10$ is an error-correcting code in the following sense:
ISBN-10 is Error-Correcting Code/Transposition Error
If any two of the first $9$ digits are transposed, the check digit will be wrong.
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.
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 $\map V {n, p}$.
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.
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: :$\map d C = \displaystyle \min_{u \mathop \in C} \map w u$ where $\map w u$ denotes the weight of $u$.
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 $2 e + 1 \le d$.
Generation of Linear Code from Standard Generator Matrix
Let $G$ be a (standard) generator matrix for a linear code. The following methods can be used to generate a linear code from $G$:
Generation of Linear Code from Standard Generator Matrix/Method 1
A linear code $C$ can be obtained from $G$ by: :considering the rows of $G$ as codewords :forming all possible linear combinations of those codewords, considering them as vectors of a vector space.
Generation of Linear Code from Standard Generator Matrix/Method 2
A linear code $C$ can be obtained from $G$ by: :taking the set $U$ of all sequences of length $k$ over $\Z_p$ and expressing them as $1 \times k$ matrices :forming all possible matrix products $u G$ for all $u \in U$.
Golay Ternary Code has Minimum Distance 5
The Golay ternary code has a minimum distance of $5$.
Golay Ternary Code Corrects 2 Errors
The Golay ternary code corrects $2$ transmission errors.
Decoding Received Word using Coset Decoding Table
Let $C$ be a linear code. Let $v$ be a received word, which may have transmission errors. To find out the transmitted codeword $u$ corresponding to $v$: :$(1): \quad$ Find $v$ in the coset decoding table for $C$. :$(2): \quad$ The corresponding transmitted codeword $u$ will be found at the top of the column where $v$ can be found.
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.
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  $w$ is a codeword of $C$.
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  they are in the same coset of $C$.
Syndrome Decoding
Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$ To decode a given vector $v$ of $\map V {n, p}$, the syndrome of $v$ can be used as follows. Create an array $T$ of $2$ column consisting of the following: :The top row contains: ::in column $1$: the zero of $C$  ::in column $2$: its syndrome. :The $r$th row subsequent contains: ::in column $1$: any element of $\map V {n, p}$ of minimum weight which is not already included in the first $r - 1$ rows ::in column $2$: its syndrome. To decode a given vector $v$ of $\map V {n, p}$: :Calculate its syndrome :Find it in column $2$ of $T$ :See what is in column $1$ of $T$, and call it $u$, say :Subtract $u$ from $v$.
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.
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.
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 $\order {C^+} = \dfrac {\order C} 2$.
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 is minimized by the following curve, known as a catenoid: :$y = C \map \cosh {\dfrac {x + C_1} C}$ Furthermore, its area is: :$A = \paren {x_1 - x_0} C \pi + \dfrac {\pi C^2} 2 \paren {\map \sinh {\dfrac {2 \paren {x_1 + C_1} } C} - \map \sinh {\dfrac {2 \paren {x_0 + C_1} } C} }$
Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2
:$\forall x \in R: \norm x_1 = \norm x_2^\alpha$
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: :$\forall y \in R: \norm y_1 > 1 \iff \norm y_2 > 1$
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: :$\forall y \in R:\norm y_1 = 1 \iff \norm y_2 = 1$
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: :$\forall z \in U_3: \map \phi z = \begin{cases} 1 & : z = 1 \\ \omega^2 & : z = \omega \\ \omega & : z = \omega^2 \end{cases}$ Then $\phi$ is a group homomorphism.
Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.3
:$\alpha = \beta$
Max is Associative
The Max operation is associative: : $\max \left({\max \left({x, y}\right), z}\right) = \max \left({x, \max \left({y, z}\right)}\right)$ Thus we are justified in writing $\max \left({x, y, z}\right)$.
Min is Associative
The Min operation is associative: : $\min \left({\min \left({x, y}\right), z}\right) = \min \left({x, \min \left({y, z}\right)}\right)$ Thus we are justified in writing $\min \left({x, y, z}\right)$.
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: :$\set {i, \map \rho i, \map {\rho^2} i, \ldots, \map {\rho^{k - 1} } i}$ Then: :$\map {\rho^k} i = i$
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.
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 $\norm {\, \cdot \,}_1$  $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_2$ Then $\forall x \in R$: :$\norm x_1 < 1 \iff \norm x_2 < 1$
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.
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: :$\mathcal B := \set {v_1, v_2, \ldots, v_n}$ Let $*: S_n \times V \to V$ be a group action of $S_n$ on $V$ defined as: :$\forall \tuple {\rho, v} \in S_n \times V: \rho * v := \lambda_1 v_{\map \rho 1} + \lambda_2 v_{\map \rho 2} + \dotsb + \lambda_n v_{\map \rho n}$ where: : $v = \lambda_1 v_1 + \lambda_2 v_2 + \dotsb + \lambda_n v_n$ Then $*$ is a group action.
Group Action of Symmetric Group on Complex Vector Space/Orbit
The orbit of an element $v \in V$ is: :$\Orb v = \displaystyle \set {w \in V: \exists \rho \in S_n: w = \sum_{k \mathop = 1}^n \lambda_k v_{\map \rho k} }$
Group Action of Symmetric Group on Complex Vector Space/Stabilizer
The stabilizer of an element $v \in V$ is: :$\Stab v = \displaystyle \set {\rho \in S_n: \sum_{k \mathop = 1}^n \lambda_k v_k = \sum_{k \mathop = 1}^n \lambda_{\map \rho k} v_k}$
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 $\norm {\, \cdot \,}_1$  $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_2$
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 as the cycle type of $x$.
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 $G$: :$e \in \map Z G$
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: :$\conjclass e = \set e$
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.
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 $5$.
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 $\set {1, 6, 11, \ldots}$
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 $G_1 \times G_2$.
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 $G_1 \times G_2$.
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 $H$.
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.
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 $D_p$.
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.
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 : :$\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$ where: :$n \cdot 1_R = \underbrace {1_R + 1_R + \dotsb + 1_R}_{\text {$n$ times} }$
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: :::$\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$
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 $\Z_7$: :$X = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \qquad Y = \begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}$
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.
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.
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.
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: ::$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^4}$ ::$\gen {x, y: x^{15} = e = y^2, y x y^{-1} = x^{11} }$
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 $4$.
Group of Order 35 is Cyclic Group
Let $G$ be a group whose order is $35$. Then $G$ is cyclic.
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.
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 $G$.
Diagonal Relation is Reflexive
The diagonal relation $\Delta_S$ on a set $S$ is a reflexive relation in $S$.
Diagonal Relation is Symmetric
The diagonal relation $\Delta_S$ on a set $S$ is a symmetric relation in $S$.
Diagonal Relation is Transitive
The diagonal relation $\Delta_S$ on a set $S$ is a transitive relation in $S$.
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.
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 $H$.
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 $C_3 \times D_5$
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} }$
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 Definition:Non-Archimedean Division Ring Norm where: :$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
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: :$\norm {x + y} \le \max \set {\norm x, \norm y} \iff \norm {x y^{-1} + 1_R} \le \max \set {\norm {x y^{-1} }, 1}$
Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2
Then for all $i$, $0 \le i \le n$: :$\norm x^i \le \max \set {\norm x^n , 1}$
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 $H$.
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 $G \times H$.
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 $D_3$: :$G \cong C_5 \times D_3$
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 $D_5$: :$G \cong C_3 \times D_5$
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 $D_3$: :$D_6 = C_2 \times D_3$
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: :$n = \displaystyle \prod_{k \mathop = 1}^r n_k$ :$\forall k \in \set {2, 3, \ldots, r}: n_k \divides n_{k - 1}$ where $\divides$ denotes divisibility.
Parallelism is Equivalence Relation
Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$. Then $\parallel$ is an equivalence relation on $S$.
Connected Equivalence Relation is Trivial
Let $S$ be a set. Let $\mathcal R$ be a relation on $S$ which is both connected and an equivalence relation. Then $\mathcal R$ is the trivial relation on $S$.
Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3
Let $\sequence {x_n}$ be the real sequence defined as $x_n = \paren {n + 1}^{1/n}$, using exponentiation. Then $\sequence {x_n}$ converges with a limit of $1$.
Characterisation of Non-Archimedean Division Ring Norms/Corollary 1
$\norm {\,\cdot\,}$ is non-Archimedean : :$\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0}} = 1$.
Characterisation of Non-Archimedean Division Ring Norms/Corollary 3
$\norm {\,\cdot\,}$ is Archimedean : :$\sup \set {\norm {n \cdot 1_R}: n \in \N_{\gt 0} } = +\infty$
Characterisation of Non-Archimedean Division Ring Norms/Corollary 2
Let $\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0} } = C < +\infty$.
Norms Equivalent to Absolute Value on Rational Numbers
Let $\alpha \in \R_{\gt 0}$. Let $\norm{\,\cdot\,}:\Q \to \R$ be the mapping defined by: :$\forall x \in \Q: \norm{x} = \size {x}^\alpha$ where $\size {x}$ is the absolute value of $x$ in $\Q$. Then: :$\norm{\,\cdot\,}$ is a norm on $\Q$  $\,\,\alpha \le 1$
Reflexive Relation on Set of Cardinality 2 is Transitive
Let $S$ be a set whose cardinality is equal to $2$: :$\card S = 2$ Let $\odot \subseteq S \times S$ be a reflexive relation on $S$. Then $\odot$ is also transitive.
Relation on Set of Cardinality 2 cannot be Non-Symmetric and Non-Transitive
Let $S$ be a set whose cardinality is equal to $2$: :$\card S = 2$ Let $\odot \subseteq S \times S$ be a relation on $S$. Then it is not possible for $\odot$ to be not symmetric and also not transitive.
Relation on Set of Cardinality 1 is Symmetric and Transitive
Let $S$ be a set whose cardinality is equal to $1$: :$\card S = 1$ Let $\odot \subseteq S \times S$ be a relation on $S$. Then $\odot$ is both symmetric and transitive.