hcju/proofwikips huggingface dataset:
Cross-Relation on Real Numbers is Equivalence Relation
Let $\R^2$ denote the cartesian plane. Let $\alpha$ denote the relation defined on $\R^2$ by: :$\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$ Then $\alpha$ is an equivalence relation on $\R^2$.
Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation
The equivalence classes of $\alpha$, when interpreted as points in the plane, are the straight lines of slope $1$.
Subtraction on Numbers is Anticommutative/Natural Numbers
The operation of subtraction on the natural numbers $\N$ is anticommutative, and defined only when $a = b$: That is: :$a - b = b - a \iff a = b$
Subtraction on Numbers is Anticommutative/Integral Domains
The operation of subtraction on the numbers is anticommutative. That is: :$a - b = b - a \iff a = b$
Natural Number Subtraction is not Closed
The operation of subtraction on the natural numbers is not closed.
Integers under Addition form Semigroup
The set of integers under addition $\struct {\Z, +}$ forms a semigroup.
Natural Numbers under Multiplication form Subsemigroup of Integers
Let $\struct {\N, \times}$ denote the set of natural numbers under multiplication. Let $\struct {\Z, \times}$ denote the set of integers under multiplication. Then $\struct {\N, \times}$ is a subsemigroup of $\struct {\Z, \times}$.
Cauchy Sequence Is Eventually Bounded Away From Non-Limit
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Suppose $\sequence {x_n}$ does not converge to $l \in R$, then: :$\exists K \in \N$ and $C \in \R_{>0}: \forall n > K: C < \norm {x_n - l}$
Embedding Division Ring into Quotient Ring of Cauchy Sequences
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$ Let $\NN = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0}$ Let $\norm {\, \cdot \,}: \CC \, \big / \NN \to \R_{\ge 0}$ be the norm on the quotient ring $\CC \, \big / \NN$ defined by: :$\displaystyle \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN} = \lim_{n \mathop \to \infty} \norm {x_n}$ Let $\phi: R \to \CC \, \big / \NN$ be the mapping from $R$ to the quotient ring $\CC \, \big / \NN$ defined by: :$\forall a \in R: \map \phi a = \sequence {a, a, a, \dotsc} + \NN$ where $\sequence {a, a, a, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {a, a, a, \dotsc}$. Then: :$\phi$ is a distance-preserving ring monomorphism.
Non-Zero Real Numbers under Multiplication form Group
Let $\R_{\ne 0}$ be the set of real numbers without zero: :$\R_{\ne 0} = \R \setminus \set 0$ The structure $\struct {\R_{\ne 0}, \times}$ forms a group.
Symmetric Group on n Letters is Isomorphic to Symmetric Group
The symmetric group on $n$ letters $\struct {S_n, \circ}$ is isomorphic to the symmetric group on the $n$ elements of any set $T$ whose cardinality is $n$. That is: :$\forall T \subseteq \mathbb U, \card T = n: \struct {S_n, \circ} \cong \struct {\Gamma \paren T, \circ}$
Symmetric Groups of Same Order are Isomorphic
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $T_1$ and $T_2$ be sets whose cardinality $\card {T_1}$ and $\card {T_2}$ are both $n$. Let $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ be the symmetric group on $S$ and $T$ respectively. Then $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ are isomorphic.
Quotient Ring of Cauchy Sequences is Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$ Let $\NN$ be the set of null sequences. For all $\sequence {x_n} \in \CC$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \NN$ Let $\norm {\, \cdot \,}_1: \CC \,\big / \NN \to \R_{\ge 0}$ be defined by: :$\displaystyle \forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = \lim_{n \mathop \to \infty} \norm {x_n}$ Then: :$\struct {\CC \,\big / \NN, \norm {\, \cdot \,}_1 }$ is a normed division ring.
Inequality Rule for Real Sequences
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$. Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ :$\displaystyle \lim_{n \mathop \to \infty} y_n = m$ Let there exist $N \in \N$ such that: :$\forall n \ge N: x_n \le y_n$ Then: :$l \le m$
Rule of Association/Disjunction/Formulation 2/Reverse Implication
:$\vdash \left({p \lor \left({q \lor r}\right)}\right) \impliedby \left({\left({p \lor q}\right) \lor r}\right)$
Left Regular Representation of Subset Product
Let $\struct {S, \circ}$ be a magma. Let $T \subseteq S$ be a subset of $S$. Let $\lambda_a: S \to S$ be the left regular representation of $S$ with respect to $a$. Then: :$\lambda_a \sqbrk T = \set a \circ T = a \circ T$ where $a \circ T$ denotes subset product with a singleton.
Right Regular Representation of Subset Product
Let $\struct {S, \circ}$ be a magma. Let $T \subseteq S$ be a subset of $S$. Let $\rho_a: S \to S$ be the right regular representation of $S$ with respect to $a$. Then: :$\rho_a \sqbrk T = T \circ \set a = T \circ a$ where $T \circ a$ denotes subset product with a singleton.
Order of Cycle is Length of Cycle
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$ be a cyclic permutation of length $k$. Then: :$\order \pi = k$ where: :$\order \pi$ denotes the order of $\pi$ in $S_n$.
Norm Sequence of Cauchy Sequence has Limit
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Then $\sequence {\norm {x_n} }$ has a limit in $\R$. That is, :$\exists l \in \R: \displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = l$
Equivalent Cauchy Sequences have Equal Limits of Norm Sequences
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ and $\sequence {y_n}$ be Cauchy sequences in $R$. Let $\displaystyle \lim_{n \mathop \to \infty} {x_n - y_n} = 0$. Then: :$\displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = \lim_{n \mathop \to \infty} \norm {y_n}$
Rule of Conjunction/Sequent Form/Formulation 2
:$\vdash p \implies \paren{ q \implies \paren{ p \land q } }$
Subgroup Generated by One Element is Cyclic
Let $G$ be a group. Let $a \in G$. Then $\gen a$, the subgroup generated by $a$, is cyclic:
Order of Elements in Quaternion Group
Let $Q = \Dic 2$ be the quaternion group, whose group presentation is given by: :$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ Then $\Dic 2$ has: : $1$ element of order $2$ and: : $6$ elements of order $4$.
Reduced Residues Modulo 5 under Multiplication form Cyclic Group
Let $\struct {\Z'_5, \times_5}$ denote the multiplicative group of reduced residues modulo $5$. Then $\struct {\Z'_5, \times_5}$ is cyclic.
Number of Generators of Cyclic Group whose Order is Power of 2
Let $G$ be a finite cyclic group. Let the order of $G$ be $2^k$ for some $k \in \Z_{>0}$. Then $G$ has $2^{n - 1}$ distinct generators.
Element of Group is in its own Coset/Left
Let: : $x H$ be the left coset of $x$ modulo $H$. Then: : $x \in x H$
Element of Group is in its own Coset/Right
Let: : $H x$ be the right coset of $x$ modulo $H$. Then: : $x \in H x$
Element of Group is in Unique Coset of Subgroup/Right
There exists a exactly one right coset of $H$ containing $x$, that is: $H x$
Equivalence of Definitions of Infinite Order Element
{{TFAE|def = Infinite Order Element}} Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$.
Equivalence of Definitions of Finite Order Element
{{TFAE|def = Finite Order Element}} Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$.
Right Cosets are Equal iff Left Cosets by Inverse are Equal
Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $g_1, g_2 \in G$. Then: :$H g_1 = H g_2 \iff {g_1}^{-1} H = {g_2}^{-1} H$ where: :${g_1}^{-1}$ and ${g_2}^{-1}$ denote the inverses of $g_1$ and $g_2$ in $G$ :$H g_1$ and $H g_2$ denote the right cosets of $H$ by $g_1$ and $g_2$ respectively :${g_1}^{-1} H$ and ${g_2}^{-1} H$ denote the left cosets of $H$ by ${g_1}^{-1}$ and ${g_2}^{-1}$ respectively.
Subgroup of Subgroup with Prime Index
Let $\struct {G, \circ}$ be a group. Let $H$ be a subgroup of $G$. Let $K$ be a subgroup of $H$. Let: :$\index G K = p$ where: :$p$ denotes a prime number :$\index G K$ denotes the index of $K$ in $G$. Then either: :$H = K$ or: :$H = G$
Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations
Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations. That is, for every WFF $\mathbf A$: :$\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathscr H_2} \mathbf A$
Klein Four-Group as Subgroup of S4
Let $G$ be the following subset of the symmetric group on $4$ letters $S_4$, expressed in two-row notation: {{begin-eqn}} {{eqn | l = e       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{bmatrix}       | c =  }} {{eqn | l = a       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{bmatrix}       | c =  }} {{eqn | l = b       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{bmatrix}       | c =  }} {{eqn | l = c       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{bmatrix}       | c =  }} {{end-eqn}} Then $G$ is an example of the Klein $4$-group.
Non-Abelian Order 10 Group has Order 5 Element
Let $G$ be a non-abelian group of order $10$. Then $G$ has at least one element of order $5$.
Quotient of Cauchy Sequences is Metric Completion
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $d$ be the metric induced by $\struct {R, \norm {\, \cdot \,} }$. Let $\mathcal C$ be the ring of Cauchy sequences over $R$. Let $\mathcal N$ be the set of null sequences in $R$. Let $\mathcal C \,\big / \mathcal N$ be the quotient ring of Cauchy sequences of $\mathcal C$ by the maximal ideal $\mathcal N$. Let $\norm {\, \cdot \,}: \mathcal C \,\big / \mathcal N \to \R_{\ge 0}$ be the norm on the quotient ring $\mathcal C \,\big / \mathcal N$ defined by: :$\displaystyle \forall \sequence {x_n} + \mathcal N: \norm {\sequence {x_n} + \mathcal N} = \lim_{n \mathop \to \infty} \norm{x_n}$ Let $d'$ be the metric induced by $\struct {\mathcal C \,\big / \mathcal N, \norm {\, \cdot \,} }$ Let $\phi: R \to \mathcal C \,\big / \mathcal N$ be the mapping from $R$ to the quotient ring $\mathcal C \,\big / \mathcal N$ defined by: :$\forall a \in R: \map \phi a = \sequence {a, a, a, \ldots} + \mathcal N$ where $\sequence {a, a, a, \ldots} + \mathcal N$ is the left coset in $\mathcal C \, \big / \mathcal N$ that contains the constant sequence $\sequence {a, a, a, \ldots}$. Then: :$\struct {\mathcal C \,\big / \mathcal N, d'}$ is the metric completion of $\struct {R,d}$ and: :$\map \phi R$ is a dense subset of $\mathcal C \,\big / \mathcal N$
Completion of Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Then: :$\struct {R, \norm {\, \cdot \,} }$ has a normed division ring completion $\struct {R', \norm {\, \cdot \,}' }$
Group of Order 27 has Subgroup of Order 3
Let $G$ be a group whose identity element is $e$. Let $G$ be of order $27$. Then $G$ has at least one subgroup of order $3$.
Group does not Necessarily have Subgroup of Order of Divisor of its Order
Let $G$ be a finite group whose order is $n$. Let $d$ be a divisor of $n$. Then it is not necessarily the case that $G$ has a subgroup of order $d$.
Characteristic Function of Normal Distribution
The characteristic function of the normal distribution with mean $\mu$ and variance $\sigma^2$ is :$\map \phi t = e^{i t \mu - \frac 1 2 t^2 \sigma^2}$
Product of Subset with Intersection/Equality does not Hold
While it is the case that: :$X \circ \paren {Y \cap Z} \subseteq \paren {X \circ Y} \cap \paren {X \circ Z}$ it is not necessarily the case that: :$X \circ \paren {Y \cap Z} = \paren {X \circ Y} \cap \paren {X \circ Z}$
Left Cosets are Equal iff Element in Other Left Coset
Let $x H$ denote the left coset of $H$ by $x$. Then: :$x H = y H \iff x \in y H$
Right Cosets are Equal iff Element in Other Right Coset
Let $H x$ denote the right coset of $H$ by $x$. Then: :$H x = H y \iff x \in H y$
Order of Element divides Order of Centralizer
Let $G$ be a finite group. Let $x \in G$ be an element of $G$. Let $\map {C_G} x$ denote the centralizer of $x$. Then: :$\order x \divides \order {\map {C_G} x}$ where: :$\order x$ denotes the order of $x$ in $G$ :$\divides$ denotes divisibility :$\order {\map {C_G} x}$ denotes the order of $\map {C_G} x$.
Left Coset of Stabilizer in Group of Transformations
Let $S$ be a non-empty set. Let $G$ be a group of permutations of $S$. Let $t \in G$. Let $G_t$ be the set defined as: :$G_t = \set {g \in G: \map g t = t}$ Then each left coset of $G_t$ in $G$ consists of the elements of $G$ that map $t$ to some element of $S$. {{explain|The source work does not discuss group actions, but still defines $G_t$ as the stabilizer of $t$ in $G$. This needs to be reviewed and put into the language of group actions as a result related to transformation group action -- but this area of group theory is not as well covered in  as it ought to be. I need to dig out my college notes on group actions, which were more comprehensive and understandable than any of the other works I have on my shelf, which will also need to be exploited properly.<br>Hence the second part of this question in Whitelaw is not covered yet.}}
Normed Division Ring is Field iff Completion is Field
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$ Then: :$R$ is a field  $R'$ is a field.
Non-Abelian Order 8 Group has Order 4 Element
Let $G$ be a non-abelian group of order $8$. Then $G$ has at least one element of order $4$.
Group of Prime Order p has p-1 Elements of Order p
Let $p$ be a prime number. Let $G$ be a group with identity $e$ whose order is $p$. Then $G$ has $p - 1$ elements of order $p$.
Number of Order p Elements in Group with m Order p Subgroups
Let $G$ be a group whose identity is $e$. Let $G$ have $m$ subgroups of order $p$. The total number of elements of $G$ of order $p$ is $m \paren {p - 1}$.
Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2
Let $p$ be a prime number. Let $G$ be a finite abelian group whose identity is $e$. Let $G$ have at least $p$ elements of order $p$. Then: : $p^2 \divides \order G$ where: :$\divides$ denotes divisibility :$\order G$ denotes the order of $G$.
Abelian Group of Semiprime Order is Cyclic
Let $p$ and $q$ be distinct prime numbers. Let $G$ be an abelian group such that: :$\order G = p q$ where $\order G$ denotes the order of $G$. Then $G$ is cyclic.
General Linear Group to Determinant is Homomorphism/Corollary
The kernel of the $\det$ mapping is the special linear group $\SL {n, \R}$.
Quotient Group of General Linear Group by Special Linear Group
Let $\GL {n, \R}$ denote the general linear group of degree $n$ over $\R$. Let $\SL {n, \R}$ denote the special linear group of degree $n$ over $\R$. Then the quotient group $\GL {n, \R} / \SL {n, \R}$ is the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.
Normed Division Ring Completions are Isometric and Isomorphic/Lemma 1
Let $\psi' = \phi_2 \circ \phi_1^{-1}:\phi_1 \paren R \to \phi_2 \paren R$ be the composition of $\phi_1^{-1}$ with $\phi_2$.  Then $\psi': \struct {\map {\phi_1} R, \norm {\, \cdot \,}_1 } \to \struct {\map {\phi_2} R, \norm {\, \cdot \,}_2 }$ is an isometric ring isomorphism.
Normed Division Ring Completions are Isometric and Isomorphic/Lemma 2
Let $\psi: S_1 \to S_2$ be defined by: :$\forall x \in S_1: \map \psi x = \displaystyle \lim_{n \mathop \to \infty} \map {\psi'} {x_n}$ where $x = \displaystyle \lim_{n \mathop \to \infty} x_n$ for some sequence $\sequence {x_n} \subseteq R_1$ Then $\psi$ is a well-defined mapping.
Normed Division Ring Completions are Isometric and Isomorphic/Lemma 3
:$\psi$ is a surjective mapping.
Normed Division Ring Completions are Isometric and Isomorphic/Lemma 4
:$\psi$ is an isometry.
Normed Division Ring Completions are Isometric and Isomorphic/Lemma 5
:$\psi$ is a ring isomorphism.
Normed Division Ring Completions are Isometric and Isomorphic
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {S_1, \norm {\, \cdot \,}_1 }$ and $\struct {S_2, \norm {\, \cdot \,}_2 }$ be normed division ring completions of $\struct {R, \norm {\, \cdot \,} }$ Then there exists an isometric isomorphism $\psi: \struct {S_1, \norm {\, \cdot \,}_1 } \to \struct {S_2, \norm {\, \cdot \,}_2 }$
Distance-Preserving Image Isometric to Domain for Metric Spaces
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $\phi: M_1 \to M_2$ be a distance-preserving mapping. Then: :$\phi: M_1 \to \Img \phi$ is an isometry.
Distance-Preserving Mapping is Injection of Metric Spaces
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $\phi: M_1 \to M_2$ be a distance-preserving mapping. Then $\phi$ is an injection.
Odd Power Function is Surjective
Let $n \in \Z_{\ge 0}$ be an odd positive integer. Let $f_n: \R \to \R$ be the real function defined as: :$\map {f_n} x = x^n$ Then $f_n$ is a surjection.
Residue at Multiple Pole
Let $f: \C \to \C$ be a function meromorphic on some region, $D$, containing $a$.  Let $f$ have a single pole in $D$, of order $N$, at $a$. Then the residue of $f$ at $a$ is given by: :$\displaystyle \Res f a = \frac 1 {\paren {N - 1}!} \lim_{z \mathop \to a} \frac { \d^{N - 1} } { \d z^{N - 1} } \paren {\paren {z - a}^N \map f z}$
Bijection from Cartesian Product of Initial Segments to Initial Segment
Let $\N_k$ be used to denote the set of the first $k$ non-zero natural numbers: :$\N_k := \set {1, 2, \ldots, k}$ Then a bijection can be established between $\N_k \times \N_l$ and $\N_{k l}$, where $\N_k \times \N_l$ denotes the Cartesian product of $\N_k$ and $\N_l$.
Bijection between S x T and T x S
Let $S$ and $T$ be sets. Let $S \times T$ be the Cartesian product of $S$ and $T$. Then there exists a bijection from $S \times T$ to $T \times S$.
Trivial Group is Group
The trivial group is a group.
Order of Dihedral Group
The dihedral group $D_n$ is of order $2 n$.
Matrix Entrywise Addition is Commutative
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems. For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is commutative on $\map \MM {m, n}$. That is: :$\mathbf A + \mathbf B = \mathbf B + \mathbf A$ for all $\mathbf A$ and $\mathbf B$ in $\map \MM {m, n}$.
Special Linear Group is not Abelian
Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\SL {n, K}$ be the special linear group of order $n$ over $K$. Then $\SL {n, K}$ is not an abelian group.
Summation Formula (Complex Analysis)/Lemma
Let $C_N$ be the square with vertices $\left({N + \frac 1 2}\right) \left({\pm 1 \pm i}\right)$ for $N \in \N$.  Then there exists a constant $A$ independent of $N$ such that:  :$\displaystyle \left\vert{\cot \left({\pi z}\right)}\right\vert < A$  for all $z$ on $C_N$.
Product of Generating Elements of Dihedral Group
Let $D_n$ be the dihedral group of order $2 n$. Let $D_n$ be defined by its group presentation: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Then for all $k \in \Z_{\ge 0}$: :$\beta \alpha^k = \alpha^{n - k} \beta$
Center of Dihedral Group
Let $n \in \N$ be a natural number such that $n \ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Let $\map Z {D_n}$ denote the center of $D_n$. Then: :$\map Z {D_n} = \begin{cases} e & : n \text { odd} \\ \set {e, \alpha^{n / 2} } & : n \text { even} \end{cases}$
Intersection of Additive Groups of Integer Multiples
Let $m, n \in \Z_{> 0}$ be (strictly) positive integers. Let $\struct {m \Z, +}$ and $\struct {n \Z, +}$ be the corresponding additive groups of integer multiples. Then: :$\struct {m \Z, +} \cap \struct {n \Z, +} = \struct {\lcm \set {m, n} \Z, +}$
Subgroup of Additive Group of Integers Generated by Two Integers
Let $m, n \in \Z_{> 0}$ be (strictly) positive integers. Let $\struct {\Z, +}$ denote the additive group of integers. Let $\gen {m, n}$ be the subgroup of $\struct {\Z, +}$ generated by $m$ and $n$. Then: :$\gen {m, n} = \struct {\gcd \set {m, n} \Z, +}$ That is, the additive groups of integer multiples of $\gcd \set {m, n}$, where $\gcd \set {m, n}$ is the greatest common divisor of $m$ and $n$.
Subgroups of Cartesian Product of Additive Group of Integers
Let $\struct {\Z, +}$ denote the additive group of integers. Let $m, n \in \Z_{> 0}$ be (strictly) positive integers. Let $\struct {\Z \times \Z, +}$ denote the Cartesian product of $\struct {\Z, +}$ with itself. The subgroups of $\struct {\Z \times \Z, +}$ are not all of the form: :$\struct {m \Z, +} \times \struct {n \Z, +}$ where $\struct {m \Z, +}$ denotes the additive group of integer multiples of $m$.
Equivalence Relation on Symmetric Group by Image of n is Congruence Modulo Subgroup
Let $S_n$ denote the symmetric group on $n$ letters $\set {1, \dots, n}$. Let $\sim$ be the relation on $S_n$ defined as: :$\forall \pi, \tau \in S_n: \pi \sim \tau \iff \map \pi n = \map \tau n$ Then $\sim$ is an equivalence relation which is congruence modulo a subgroup. {{explain|Work needed to be done to explain exactly what is happening here.}}
Order of Multiplicative Group of Reduced Residues
Let $\struct {\Z'_m, \times_m}$ denote the multiplicative group of reduced residues modulo $m$. The order of $\struct {\Z'_m, \times_m}$ is $\map \phi m$, where $\phi$ denotes the Euler $\phi$ function.
Circle Group is Uncountably Infinite
The circle group $\struct {K, \times}$ is an uncountably infinite group.
Subgroups of Symmetry Group of Regular Hexagon
Let $\mathcal H = ABCDEF$ be a regular hexagon. Let $D_6$ denote the symmetry group of $\mathcal H$. :520px Let $e$ denote the identity mapping Let $\alpha$ denote rotation of $\mathcal H$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$) Let $\beta$ denote reflection of $\mathcal H$ in the $AD$ axis. The subsets of $D_6$ which form its subgroups are as follows: ;Order $1$: :$\set e$ ;Order $2$: :$\set {e, \alpha^3}$ :$\set {e, \beta}$ :$\set {e, \alpha \beta}$ :$\set {e, \alpha^2 \beta}$ :$\set {e, \alpha^3 \beta}$ :$\set {e, \alpha^4 \beta}$ :$\set {e, \alpha^5 \beta}$ ;Order $3$: :$\set {e, \alpha^2, \alpha^4}$ ;Order $4$: :$\set {e, \alpha^3, \beta, \alpha^3 \beta}$ :$\set {e, \alpha^3, \alpha \beta, \alpha^4 \beta}$ :$\set {e, \alpha^3, \alpha^2 \beta, \alpha^5 \beta}$ ;Order $6$: :$\set {e, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5}$ :$\set {e, \alpha^2, \alpha^4, \beta, \alpha^2 \beta, \alpha^4 \beta}$ :$\set {e, \alpha^2, \alpha^4, \alpha \beta, \alpha^3 \beta, \alpha^5 \beta}$ ;Order $12$: :$D_6$ itself.
Subgroup of Symmetric Group that Fixes n
Let $S_n$ denote the symmetric group on $n$ letters. Let $H$ denote the subgroup of $S_n$ which consists of all $\pi \in S_n$ such that: :$\map \pi n = n$ Then: :$H = S_{n - 1}$ and the index of $H$ in $S_n$ is given by: :$\index {S_n} H = n$
Product of Orders of Abelian Group Elements Divides LCM of Order of Product
Let $G$ be an abelian group. Let $a, b \in G$. Then: :$\order {a b} \divides \lcm \set {\order a, \order b}$ where: :$\order a$ denotes the order of $a$ :$\divides$ denotes divisibility :$\lcm$ denotes the lowest common multiple.
Groups of Order 4
There exist exactly $2$ groups of order $4$, up to isomorphism: :$C_4$, the cyclic group of order $4$ :$K_4$, the Klein $4$-group.