hcju/proofwikips huggingface dataset:
Difference of Two Even Powers
Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then for all $a, b \in \mathbb F$: {{begin-eqn}} {{eqn | l = a^{2 n} - b^{2 n}       | r = \paren {a - b} \paren {a + b} \sum_{j \mathop = 0}^{n - 1} a^{2 \paren {n - j - 1} } b^{2 j}       | c =  }} {{eqn | r = \paren {a - b} \paren {a + b} \paren {a^{2 n - 2} + a^{2 n - 4} b^2 + a^{2 n - 6} b^4 + \dotsb + a^2 b^{2 n - 4} + b^{2 n - 2} }       | c =  }} {{end-eqn}}
Factors of Difference of Two Odd Powers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = x^{2 n + 1} - y^{2 n + 1}       | r = \paren {x - y} \displaystyle \prod_{k \mathop = 1}^n \paren {x^2 - 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}       | c =  }} {{eqn | r = \paren {x - y} \paren {x^2 - 2 x y \cos \dfrac {2 \pi} {2 n + 1} + y^2} \paren {x^2 - 2 x y \cos \dfrac {4 \pi} {2 n + 1} + y^2} \dotsm \paren {x^2 - 2 x y \cos \dfrac {2 n \pi} {2 n + 1} + y^2}       | c =  }} {{end-eqn}}
Factors of Sum of Two Odd Powers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = x^{2 n + 1} + y^{2 n + 1}       | r = \paren {x + y} \displaystyle \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}       | c =  }} {{eqn | r = \paren {x + y} \paren {x^2 + 2 x y \cos \dfrac {2 \pi} {2 n + 1} + y^2} \paren {x^2 + 2 x y \cos \dfrac {4 \pi} {2 n + 1} + y^2} \dotsm \paren {x^2 + 2 x y \cos \dfrac {2 n \pi} {2 n + 1} + y^2}       | c =  }} {{end-eqn}}
Factors of Difference of Two Even Powers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \displaystyle \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$
Factors of Sum of Two Even Powers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$x^{2 n} + y^{2 n} = \displaystyle \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {\paren {2 k - 1} \pi} {2 n} + y^2}$
Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen
:The $r$-sphere of $x$, $\map {S_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.
Magnitude of Projection of Complex Number on Another
Let $z_1$ and $z_2$ denote complex numbers in vector form. Let $\map {\pr_1} {z_1, z_2}$ denote the projection of $z_1$ on $z_2$. {{explain|We really need another page to explain the concept of Definition:Projection (Analytic Geometry) in the context of the Definition:Complex Plane}} Then: :$\cmod {\map {\pr_1} {z_1, z_2} } = \dfrac {\cmod {z_1 \circ z_2} } {\cmod {z_2} }$ where: :$z_1 \circ z_2$ denotes complex dot product :$\cmod {z_2}$ denotes complex modulus.
Non-Archimedean Division Ring is Totally Disconnected
Let $\struct {R, \norm{\,\cdot\,} }$ be a non-Archimedean normed division ring. Let $\tau$ be the topology induced by the norm $\norm{\,\cdot\,}$. Then the topological space $\struct {R, \tau}$ is totally disconnected.
Distance between Points in Complex Plane
Let $A$ and $B$ be points in the complex plane such that: :$A = \tuple {x_1, y_1}$ :$B = \tuple {x_2, y_2}$ Then the distance between $A$ and $B$ is given by: {{begin-eqn}} {{eqn | l = \size {AB}       | r = \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}       | c =  }} {{eqn | r = \cmod {z_1 - z_2}       | c =  }} {{end-eqn}} where $z_1$ and $z_2$ are represented by the complex numbers $z_1$ and $z_2$ respectively.
Linear Combination of Non-Parallel Complex Numbers is Zero if Factors are Both Zero
Let $z_1$ and $z_2$ be complex numbers expressed as vectors such taht $z_1$ is not parallel to $z_2$. Let $a, b \in \R$ be real numbers such that: :$a z_1 + b z_2 = 0$ Then $a = 0$ and $b = 0$.
Equation for Line through Two Points in Complex Plane/Formulation 1
$L$ can be expressed by the equation: :$\map \arg {\dfrac {z - z_1} {z_2 - z_1} } = 0$
Equation for Line through Two Points in Complex Plane/Parametric Form 1
$L$ can be expressed by the equation: :$z = z_1 + t \paren {z_2 - z_1}$ or: :$z = \paren {1 - t} z_1 + t z_2$ This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''.
Equation for Line through Two Points in Complex Plane/Symmetric Form
$L$ can be expressed by the equation: :$z = \dfrac {m z_1 + n z_2} {m + n}$ This form of $L$ is known as the '''symmetric form'''.
Valuation Ring of Non-Archimedean Division Ring is Subring
Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$. Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is: :$\OO = \set {x \in R : \norm{x} \le 1}$ Then $\OO$ is a subring of $R$: :with a unity: $1_R$ :in which there are no (proper) zero divisors, that is: :::$\forall x, y \in \OO: x \circ y = 0_R \implies x = 0_R \text{ or } y = 0_R$
Valuation Ideal is Maximal Ideal of Induced Valuation Ring
Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$. Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is: :$\OO = \set{x \in R : \norm x \le 1}$ Let $\PP$ be the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is: :$\PP = \set{x \in R : \norm x < 1}$ Then $\PP$ is an ideal of $\OO$: :$(a):\quad \PP$ is a maximal left ideal :$(b):\quad \PP$ is a maximal right ideal :$(c):\quad$ the quotient ring $\OO / \PP$ is a division ring.
Sequence of Imaginary Reciprocals
Consider the subset $S$ of the complex plane defined as: :$S := \set {\dfrac i n : n \in \Z_{>0} }$ That is: :$S := \set {i, \dfrac i 2, \dfrac i 3, \dfrac i 4, \ldots}$ where $i$ is the imaginary unit.
Sequence of Imaginary Reciprocals/Boundedness
The set $S$ is bounded in $\C$.
Sequence of Imaginary Reciprocals/Limit Points
The set $S$ has exactly one limit point, and that is $z = 0$.
Sequence of Imaginary Reciprocals/Closedness
The set $S$ is not closed.
Valuation Ring of P-adic Norm on Rationals
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. The induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set: :$\OO = \Z_{\paren p} = \set {\dfrac a b \in \Q : p \nmid b}$
Valuation Ideal of P-adic Norm on Rationals
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. The induced valuation ideal on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set: :$\PP = p \Z_{\ideal p} = \set {\dfrac a b \in \Q : p \nmid b, p \divides a}$ where $\Z_{\ideal p}$ is the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$
Residue Field of P-adic Norm on Rationals
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. The induced residue field on $\struct {\Q,\norm {\,\cdot\,}_p}$ is isomorphic to the field $\F_p$ of integers modulo $p$.
Sequence of Imaginary Reciprocals/Boundary Points
Every point of $S$, along with the point $z = 0$, is a boundary point of $S$.
Sequence of Imaginary Reciprocals/Interior
No point of $S$ is an interior point.
Sequence of Imaginary Reciprocals/Openness
$S$ is not an open set.
Sequence of Imaginary Reciprocals/Connectedness
$S$ is not connected.
Sequence of Imaginary Reciprocals/Not an Open Region
$S$ is not an open region.
Sequence of Imaginary Reciprocals/Not Compact
The set $S$ is not compact.
Sequence of Imaginary Reciprocals/Closure is Compact
The closure $S^-$ of the set $S$ is compact.
Product of Complex Conjugates/Examples/3 Arguments
Let $z_1, z_2, z_3 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 z_2 z_3} = \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}$
Residue Field of P-adic Norm on Rationals/Lemma 1
:$\phi$ is a homomorphism.
Residue Field of P-adic Norm on Rationals/Lemma 2
:$p \Z = \map \ker \phi$
Residue Field of P-adic Norm on Rationals/Lemma 3
:$\phi : \Z \to \Z_{\paren p} / p \Z_{\paren p}$ is a surjection.
Equation for Line through Two Points in Complex Plane/Parametric Form 2
$L$ can be expressed by the equations: {{begin-eqn}} {{eqn | l = x - x_1       | r = t \paren {x_2 - x_1} }} {{eqn | l = y - y_1       | r = t \paren {y_2 - y_1} }} {{end-eqn}} These are the '''parametric equations of $L$''', where $t$ is the parameter.
Equation for Perpendicular Bisector of Two Points in Complex Plane/Parametric Form 1
$L$ can be expressed by the equation: :$z = ...$ or: :$z = ...$ This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''.
Geodesic Equation/2d Surface Embedded in 3d Euclidean Space
Let $\sigma: U \subset \R^2 \to V \subset \R^3$ be a smooth surface specified by a vector-valued function: :$\mathbf r = \map {\mathbf r} {u, v}$ Then a geodesic of $\sigma$ satisfies the following system of differential equations: :$\dfrac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {E u' + F v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$ :$\dfrac {E_v u'^2 + 2 F_v u' v' + G_v v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {F u' + G v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$ where $E, F, G$ are the functions of the first fundamental form: :$\displaystyle E = {\mathbf r}_u \cdot {\mathbf r}_u, F = {\mathbf r}_u \cdot {\mathbf r}_v, G = {\mathbf r}_v \cdot {\mathbf r}_v$
Modulus of Exponential of i z where z is on Circle
Let $C$ be the circle embedded in the complex plane given by the equation: :$z = R e^{i \theta}$ Then: :$\cmod {e^{i z} } = e^{-R \sin \theta}$
Sum of Complex Numbers in Exponential Form/General Result
Let $n \in \Z_{>0}$ be a positive integer. For all $k \in \set {1, 2, \dotsc, n}$, let: :$z_k = r_k e^{i \theta_k}$ be non-zero complex numbers in exponential form. Let: :$r e^{i \theta} = \displaystyle \sum_{k \mathop = 1}^n z_k = z_1 + z_2 + \dotsb + z_k$ Then: {{begin-eqn}} {{eqn | l = r       | r = \sqrt {\displaystyle \sum_{k \mathop = 1}^n r_k + \displaystyle \sum_{1 \mathop \le j \mathop < k \mathop \le n} 2 {r_j} {r_k} \, \map \cos {\theta_j - \theta_k} } }} {{eqn | l = \theta       | r = \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2 + \dotsb + r_n \sin \theta_n} {r_1 \cos \theta_1 + r_2 \cos \theta_2 + \dotsb + r_n \cos \theta_n} } }} {{end-eqn}}
Bias of Sample Variance
Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Let:  :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then: :$\displaystyle \hat {\sigma^2} = \frac 1 n \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$ is a biased estimator of $\sigma^2$, with:  :$\displaystyle \operatorname{bias} \paren {\hat {\sigma ^2}} = -\frac {\sigma^2} n$
Sample Mean is Unbiased Estimator of Population Mean
Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Then:  :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ is an unbiased estimator of $\mu$.
Ostrowski's Theorem/Archimedean Norm
Let $\norm {\, \cdot \,}$ be a non-trivial Archimedean norm on the rational numbers $\Q$. Then $\norm {\, \cdot \,}$ is equivalent to the absolute value $\size {\, \cdot \,}$.
Ostrowski's Theorem/Non-Archimedean Norm
Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$. Then $\norm {\, \cdot \,}$ is equivalent to the $p$-adic norm $\norm {\, \cdot \,}_p$ for some prime $p$.
Ostrowski's Theorem/Archimedean Norm/Lemma 1.1
:$\forall n \in N: \norm n \le n^\alpha$
Ostrowski's Theorem/Archimedean Norm/Lemma 1.2
:$\forall n \in N: \norm n \ge n^\alpha$
Equivalent Norms on Rational Numbers
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$. Then $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent : :$\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$
Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1
:$\exists n \in \N: 0 < \norm n < 1$.
Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2
:$n_0$ is a prime number.
Sum of Cosines of k pi over 5
:$\cos 36 \degrees + \cos 72 \degrees + \cos 108 \degrees + \cos 144 \degrees = 0$
Products of nth Roots of Unity taken up to n-1 at a Time is Zero
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $U_n = \set {z \in \C: z^n = 1}$ be the complex $n$th roots of unity. Then the sum of the products of the elements of $U_n$ taken $2, 3, 4, \dotsc n - 1$ at a time is zero.
Absolute Value of Complex Dot Product is Commutative
Let $z_1$ and $z_2$ be complex numbers. Let $z_1 \circ z_2$ denote the (complex) dot product of $z_1$ and $z_2$. Then: :$\size {z_1 \circ z_2} = \size {z_2 \circ z_1}$ where $\size {\, \cdot \,}$ denotes the absolute value function.
Three Points in Ultrametric Space have Two Equal Distances/Corollary 5
Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$. Let $a, b \in \Z_{\ne 0}$ be coprime, $a \perp b$ Then: :$\norm a = 1$ or $\norm b = 1$
Exponential of 2 m i Arccotangent of p
:$\map \exp {2 m i \arccot p} \paren {\dfrac {p i + 1} {p i - 1} }^m = 1$
Modulus z - 1 Less than Modulus z + 1 iff Real z Greater than Zero
Let $z \in \C$ be a complex number. Then: :$\cmod {z - 1} < \cmod {z + 1} \iff \map \Re z > 0$
Vertices of Equilateral Triangle in Complex Plane/Sufficient Condition
Let $z_1$, $z_2$ and $z_3$ be complex numbers. Let $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle. Then: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
Vertices of Equilateral Triangle in Complex Plane/Necessary Condition
Let $z_1$, $z_2$ and $z_3$ be complex numbers. Let $z_1$, $z_2$ and $z_3$ fulfil the condition: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$ Then $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle.
Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder
Let $\sigma$ be the surface of a cylinder. Let $\sigma$ be embedded in 3-dimensional Euclidean space. Let $\sigma$ be parameterised by $\tuple {\phi, z}$ as :$\mathbf r = \tuple {a \cos \phi, a \sin \phi, z}$ where :$a > 0$ and :$z, \phi \in \R$ Then geodesics on $\sigma$ are of the following form: :$z = C_1 \phi + C_2$ where $C_1, C_2$ are real arbitrary constants.
Modulus of Sum equals Modulus of Distance implies Quotient is Imaginary
Let $z_1$ and $z_2$ be complex numbers such that: :$\cmod {z_1 + z_2} = \cmod {z_1 - z_2}$ Then $\dfrac {z_2} {z_1}$ is wholly imaginary.
Difference of Even Powers of z + a and z - a
Let $m \in \Z$ be an integer such that $m > 1$. Then for all complex number $z$: :$\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \displaystyle \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$
Imaginary Part of Complex Product
Let $z_1$ and $z_2$ be complex numbers. Then: :$\map \Im {z_1 z_2} = \map \Re {z_1} \, \map \Im {z_2} + \map \Im {z_1} \, \map \Re {z_2}$
Element of Center in Group whose Order is Power of 2
Let $n \in \Z$ be an integer such that $n \ge 2$. Let $G$ be a group whose order is $2^n$. Let $x \in G$ be of order $2^{n - 1}$ in $G$. Then $x^{2^{n - 2} }$ is an element of the center of $G$.
Intersection of Abelian Subgroups is Normal Subgroup of Subgroup Generated by those Subgroups
Let $G$ be a group. Let $L$ and $M$ be abelian subgroups of $G$. Let $H = \gen {L, M}$ be the subgroup of $G$ generated by $L$ and $M$. Then $L \cap M$ is a normal subgroup of $H$.
Characterisation of Non-Archimedean Division Ring Norms/Corollary 5
If $\norm {\, \cdot \,}$ is non-Archimedean then: :$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = 1$ where $n \cdot 1_R =  \begin{cases} \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} } & : n > 0 \\ 0 & : n = 0 \\ \\ -\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{\text {$-n$ times} } & : n < 0 \\ \end{cases}$
Subgroup Containing all Squares of Group Elements is Normal
Let $G$ be a group. Let $H$ be a subgroup of $G$ with the property that: :$\forall x \in G: x^2 \in H$ Then $H$ is normal in $G$.
Commutator is Identity iff Elements Commute
Let $G$ be a group whose identity is $e$. Let $x, y \in G$. Let $\sqbrk {x, y}$ denote the commutator of $x$ and $y$. Then $\sqbrk {x, y} = e$  $x$ and $y$ commute.
Commutator of Quotient Group Elements
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $\sqbrk {x, y}$ denote the commutator of $x, y \in G$: :$\sqbrk {x, y} = x^{-1} y^{-1} x y$ Then: :$\forall x, y \in G: \sqbrk {x N, y N} = \sqbrk {x, y} N$ where $x N$ and $y N$ are left cosets of $N$, and so elements of the quotient group $G / N$ of $G$ by $N$.
Quotient Group is Abelian iff All Commutators in Divisor
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Then the quotient group $G / N$ is abelian : :$\forall x, y \in G: \sqbrk {x, y} \in N$ where $\sqbrk {x, y}$ denotes the commutator of $x$ and $y$.
Sufficient Condition for Quotient Group by Intersection to be Abelian
Let $G$ be a group. Let $N$ and $K$ be normal subgroups of $G$. Let the quotient groups $G / N$ and $G / K$ be abelian. Then the quotient group $G / \paren {N \cap K}$ is also abelian.
Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are
Let $G$ be a group. Let $N$ and $K$ be normal subgroups of $G$. Let the quotient groups $G / N$ and $G / K$ be cyclic. Then the quotient group $G / \paren {N \cap K}$ is not necessarily cyclic.
Order of Boolean Group is Power of 2
Let $G$ be a Boolean group. Let $\order G$ denote the order of $G$. Then: :$\order G = 2^n$ where $n \in \Z_{\ge 0}$ is a positive integer.
Subgroup of Index 2 contains all Squares of Group Elements
Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index is $2$. Then: :$\forall x \in G: x^2 \in H$
Center of Non-Abelian Group of Order pq is Trivial
Let $p$ and $q$ be distinct prime numbers. Let $G$ be a non-abelian group of order $p q$ whose identity is $e$. Then the center of $G$ is trivial: :$\map Z G = \set e$
Number of Elements of Order p in Group of Order pq is Multiple of q
Let $p$ and $q$ be distinct prime numbers. Let $G$ be a non-abelian group of order $p q$. Then the number of elements of $G$ of order $p$ is a multiple of $q$.
Intersection of Normal Subgroup with Center in p-Group
Let $p$ be a prime number Let $G$ be a $p$-group. Let $N$ be a non-trivial normal subgroup of $G$. Let $\map Z G$ denote the center of $G$. Then: :$N \cap \map Z G$ is a non-trivial normal subgroup of $G$.
Normal Subgroup of p-Group of Order p is Subset of Center
Let $p$ be a prime number. Let $G$ be a $p$-group. Let $N$ be a normal subgroup of $G$ of order $p$. Then: :$N \subseteq \map Z G$ where $\map Z G$ denotes the center of $G$.
Non-Abelian Group of Order p Cubed has Exactly One Normal Subgroup of Order p
Let $p$ be a prime number. Let $G$ be a non-abelian group of order $p^3$. Then $G$ contains exactly one normal subgroup of order $p$.
Non-Abelian Order 2p Group has Order p Element
Let $p$ be an odd prime. Let $G$ be a non-abelian group of order $2 p$. Then $G$ has at least one element of order $p$.
Fourth Power Modulo 5
Let $n \in \Z$ be an integer. Then: :$n^4 \equiv m \pmod 5$ where $m \in \set {0, 1}$.
Subgroup of Order p in Group of Order 2p is Normal/Corollary
Let $G$ be non-abelian. Every element of $G \setminus K$ is of order $2$, and: :$\forall b \in G \setminus K: b a b^{-1} = a^{-1}$
Subgroup of Order p in Group of Order 2p is Normal
Let $p$ be an odd prime. Let $G$ be a group of order $2 p$. Let $a \in G$ be of order $p$. Let $K = \gen a$ be the subgroup of $G$ generated by $a$. Then $K$ is normal in $G$.
Inner Automorphisms form Subgroup of Automorphism Group
Let $G$ be a group. Then the set $\Inn G$ of all inner automorphisms of $G$ forms a normal subgroup of the automorphism group $\Aut G$ of $G$: :$\Inn G \le \Aut G$
Property of Group Automorphism which Fixes Identity Only
Let $G$ be a finite group whose identity is $e$. Let $\phi: G \to G$ be a group automorphism. Let $\phi$ have the property that: :$\forall g \in G \setminus \set e: \map \phi t \ne t$ That is, the only fixed element of $\phi$ is $e$. Then: :$\forall x, y \in G: x^{-1} \, \map \phi x = y^{-1} \, \map \phi y \implies x = y$
Property of Group Automorphism which Fixes Identity Only/Corollary 1
:$\forall x \in G: \exists g \in G: x = g^{-1} \, \map \phi g$
Square Order 2 Matrices over Real Numbers form Ring with Unity
Let $S$ denote the set of square matrices of order $2$ whose entries are the set of real numbers. Then $S$ forms a ring with unity whose unity is the matrix $\begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$.
Product Formula for Norms on Non-zero Rationals
Let $\Q_{\ne 0}$ be the set of non-zero rational numbers. Let  $\Bbb P$ denote the set of prime numbers. Let $a \in \Q_{\ne 0}$. Then the following infinite product converges: :$\size a \times \displaystyle \prod_{p \mathop \in \Bbb P}^{} \norm a_p = 1$ where: :$\size {\,\cdot\,}$ is the absolute value on $\Q$ :$\norm {\,\cdot\,}_p$ is the $p$-adic norm on $\Q$ for prime number $p$
Null Ring is Subring of Ring
Let $R$ be a ring. Then the null ring is a subring of $R$.
Integers form Subring of Reals
The ring of integers $\struct {\Z, +, \times}$ forms a subring of the field of real numbers.
Unity of Subfield is Unity of Field
The unity of $\struct {K, +, \times}$ is also $1$.
Zero of Subring is Zero of Ring
The zero of $\struct {S, +, \times}$ is also $0$.
Zero of Subfield is Zero of Field/Proof 1
Let $\struct {F, +, \times}$ be a field whose zero is $0$. Let $\struct {K, +, \times}$ be a subfield of $\struct {F, +, \times}$. {{:Zero of Subfield is Zero of Field}}
Zero of Subfield is Zero of Field/Proof 2
Let $\struct {F, +, \times}$ be a field whose zero is $0$. Let $\struct {K, +, \times}$ be a subfield of $\struct {F, +, \times}$. {{:Zero of Subfield is Zero of Field}}
P-adic Norm not Complete on Rational Numbers/Proof 1/Case 1
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p > 3$.   Then: :$\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete normed division ring. That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.
P-adic Norm not Complete on Rational Numbers/Proof 1/Case 2
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for $p = 2$ or $3$.   Then: :$\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete normed division ring. That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.
Element in Integral Domain is Divisor iff Principal Ideal is Superset
:$x \divides y \iff \ideal y \subseteq \ideal x$
Residue of Gamma Function
Let $\Gamma$ be the Definition:Gamma Function.  Let $n$ be a non-negative integer. Then:  :$\Res \Gamma {-n} = \dfrac {\paren {-1}^n} {n!}$
Periodic Element is Multiple of Period
Let $f: \R \to \R$ be a real periodic function with period $P$. Let $L$ be a periodic element of $f$. Then $P \divides L$.
Equivalence of Definitions of Associate in Integral Domain
{{TFAE|def = Associate in Integral Domain|context = Integral Domain|view = Associate}} Let $\struct {D, +, \circ}$ be an integral domain. Let $x, y \in D$.
Finite Set of Elements in Principal Ideal Domain has GCD
Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $a_1, a_2, \dotsc, a_n$ be non-zero elements of $D$. Then $a_1, a_2, \dotsc, a_n$ all have a greatest common divisor.
Set of Linear Combinations of Finite Set of Elements of Principal Ideal Domain is Principal Ideal
Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $a_1, a_2, \dotsc, a_n$ be non-zero elements of $D$. Let $J$ be the set of all linear combinations in $D$ of $\set {a_1, a_2, \dotsc, a_n}$ Then for some $x \in D$: :$J = \ideal x$ where $\ideal x$ denotes the principal ideal generated by $x$.
Greatest Common Divisors in Principal Ideal Domain are Associates
Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $S = \set {a_1, a_2, \dotsc, a_n}$ be a set of non-zero elements of $D$. Let $y_1$ and $y_2$ be greatest common divisors of $S$. Then $y_1$ and $y_2$ are associates.