hcju/proofwikips huggingface dataset:
Conjugate of Set with Inverse Closed for Inverses
Let $G$ be a group. Let $S \subseteq G$. Let $\hat S = S \cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Let $\tilde S = \set {a s a^{-1}: s \in \hat S, a \in G}$. That is, $\tilde S$ is the set containing all the conjugates of the elements of $S$ and all their inverses. Then: : $\forall x \in \tilde S: x^{-1} \in \tilde S$
Conjugate of Set with Inverse is Closed
Let $G$ be a group. Let $S \subseteq G$. Let $\hat S = S \cup S$. Let $\tilde S = \left\{{a s a^{-1}: s \in \hat S, a \in G}\right\}$. Let $W \left({\tilde S}\right)$ be the set of words of $\tilde S$. Then $\forall w \in W \left({\tilde S}\right): \forall a \in G: a w a^{-1} \in W \left({\tilde S}\right)$.
Generator of Normal Subgroup
Let $G$ be a group. Let $S \subseteq G$. Let $\hat S = S \cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Let $\tilde S = \left\{{a s a^{-1}: s \in \hat S, a \in G}\right\}$. Let $W \left({\tilde S}\right)$ be the set of words of $\tilde S$. Let $N$ be the smallest normal subgroup of $G$ that contains $S$. Then $N = \left \langle {S} \right \rangle = W \left({\tilde S}\right)$.
Subset Product with Normal Subgroup as Generator
Let $G$ be a group whose identity is $e$. Let: :$H$ be a subgroup of $G$ :$N$ be a normal subgroup of $G$. Then: :$N \lhd \gen {N, H} = N H = H N \le G$ where: :$\le$ denotes subgroup :$\lhd$ denotes normal subgroup :$\gen {N, H}$ denotes a subgroup generator :$N H$ denotes subset product.
Subset Product of Normal Subgroups is Normal
Let $\struct {G, \circ}$ be a group. Let $N$ and $N'$ be normal subgroups of $G$. Then $N N'$ is also a normal subgroup of $G$.
Prime Group is Simple
Groups of prime order are simple.
Prime Group has no Proper Subgroups
A nontrivial group $G$ has no proper subgroups except the trivial group  $G$ is finite and its order is prime.
Quotient Group of Cyclic Group
Let $G$ be a cyclic group which is generated by $g$. Let $H$ be a subgroup of $G$. Then $g H$ generates $G / H$.
Order of Element in Quotient Group
Let $G$ be a group, and let $H$ be a normal subgroup of $G$. Let $G / H$ be the quotient group of $G$ by $H$. The order of $a H \in G / H$ divides the order of $a \in G$.
Quotient of Group by Center Cyclic implies Abelian
Let $G$ be a group. Let $\map Z G$ be the center of $G$. Let $G / \map Z G$ be the quotient group of $G$ by $\map Z G$. Let $G / \map Z G$ be cyclic. Then $G$ is abelian, so $G = \map Z G$. That is, the group $G / \map Z G$ cannot be a cyclic group which is non-trivial.
Centralizer is Normal Subgroup of Normalizer
Let $G$ be a group. Let $H \le G$ be a subgroup of $G$.  Let $\map {C_G} H$ be the centralizer of $H$ in $G$. Let $\map {N_G} H$ be the normalizer of $H$ in $G$. Let $\Aut H$ be the automorphism group of $H$. Then: :$(1): \quad \map {C_G} H \lhd \map {N_G} H$ :$(2): \quad \map {N_G} H / \map {C_G} H \cong K$ where: :$\map {N_G} H / \map {C_G} H$ is the quotient group of $\map {N_G} H$ by $\map {C_G} H$ :$K$ is a subgroup of $\Aut H$.
Number of Distinct Conjugate Subsets is Index of Normalizer
Let $G$ be a group. Let $S$ be a subset of $G$. Let $\map {N_G} S$ be the normalizer of $S$ in $G$. Let $\index G {\map {N_G} S}$ be the index of $\map {N_G} S$ in $G$. The number of distinct subsets of $G$ which are conjugates of $S \subseteq G$ is $\index G {\map {N_G} S}$.
Element of Group Not Conjugate to Proper Subgroup
Let $G$ be a finite group. Let $H$ be a proper subgroup of $G$. Then there is at least one element of $G$ not contained in $H$ or in any of its conjugates.
Second Isomorphism Theorem
=== Groups === {{:Second Isomorphism Theorem/Groups}} === Rings === {{:Second Isomorphism Theorem/Rings}} This result is also referred to by some sources as the '''first isomorphism theorem'''.
Third Isomorphism Theorem
=== Groups === {{:Third Isomorphism Theorem/Groups}} === Rings === {{:Third Isomorphism Theorem/Rings}}
Inverse of Inner Automorphism
Let $G$ be a group. Let $x \in G$. Let $\kappa_x$ be the inner automorphism of $G$ given by $x$. Then: : $\paren {\kappa_x}^{-1} = \kappa_{x^{-1} }$
Conjugates of Elements in Centralizer
Let $G$ be a group. Let $\map {C_G} a$ be the centralizer of $a$ in $G$. Then $\forall g, h \in G: g a g^{-1} = h a h^{-1}$  $g$ and $h$ belong to the same left coset of $\map {C_G} a$.
Number of Conjugates is Number of Cosets of Centralizer
Let $G$ be a group. Let $\map {C_G} a$ be the centralizer of $a$ in $G$. Then the number of different conjugates of $a$ in $G$ equals the number of different (left) cosets of $\map {C_G} a$: :$\card {\conjclass a} = \index G {\map {C_G} a}$ where: :$\conjclass a$ is the conjugacy class of $a$ in $G$ :$\index G {\map {C_G} a}$ is the index of $\map {C_G} a$ in $G$. Consequently: :$\card {\conjclass a} \divides \order G$
Size of Conjugacy Class is Index of Normalizer
Let $G$ be a group. Let $x \in G$. Let $\conjclass x$ be the conjugacy class of $x$ in $G$. Let $\map {N_G} x$ be the normalizer of $x$ in $G$. Let $\index G {\map {N_G} x}$ be the index of $\map {N_G} x$ in $G$. The number of elements in $\conjclass x$ is $\index G {\map {N_G} x}$.
Conjugacy Class of Element of Center is Singleton
Let $G$ be a group. Let $\map Z G$ denote the center of $G$. The elements of $\map Z G$ form singleton conjugacy classes, and the elements of $G \setminus \map Z G$ belong to multi-element conjugacy classes.
Conjugacy Class Equation
Let $G$ be a group. Let $\order G$ denote the order of $G$. Let $\map Z G$ denote the center of $G$. Let $x \in G$. Let $\map {N_G} x$ denote the normalizer of $x$ in $G$. Let $\index G {\map {N_G} x}$ denote the index of $\map {N_G} x$ in $G$. Let $m$ be the number of non-singleton conjugacy classes of $G$. Let $x_j: j \in \set {1, 2, \ldots, m}$ be arbitrary elements of those conjugacy classes. Then: :$\displaystyle \order G = \order {\map Z G} + \sum_{j \mathop = 1}^m \index G {\map {N_G} {x_j} }$
Group of Order Prime Squared is Abelian
A group whose order is the square of a prime is abelian.
Center of Group of Prime Power Order is Non-Trivial
Let $G$ be a group whose order is the power of a prime. Then the center of $G$ is non-trivial: :$\forall G: \order G = p^r: p \in \mathbb P, r \in \N_{>0}: \map Z G \ne \set e$
Center of Group of Order Prime Cubed
Let $G$ be a group of order $p^3$, where $p$ is a prime. Let $\map Z G$ be the center of $G$. Then $\order {\map Z G} \ne p^2$.
Prime Power Group has Non-Trivial Proper Normal Subgroup
Let $G$ be a group, whose identity is $e$, such that $\order G = p^n: n > 1, p \in \mathbb P$. Then $G$ has a proper normal subgroup which is non-trivial.
Composition Series of Group of Prime Power Order
Let $G$ be a group whose identity is $e$, and whose order is a prime power: :$\order G = p^n, p \in \mathbb P, n \ge 1$ Then $G$ has a composition series: :$\set e = G_0 \subset G_1 \subset \ldots \subset G_n = G$ such that $\order {G_k} = p^k$, $G_k \lhd G_{k + 1}$ and $G_{k + 1} / G_k$ is cyclic and of order $p$.
Sum Rule for Counting
Let there be: : $r_1$ different objects in the set $S_1$ : $r_2$ different objects in the set $S_2$ : $\ldots$ : $r_m$ different objects in the set $S_m$. Let $\displaystyle \bigcap_{i \mathop = 1}^m S_i = \varnothing$. Then the number of ways to select an object from one of the $m$ sets is $\displaystyle \sum_{i \mathop = 1}^m r_i$.
Product Rule for Counting
Let it be possible to choose an element $\alpha$ from a given set $S$ in $m$ different ways. Let it be possible to choose an element $\beta$ from a given set $T$ in $n$ different ways. Then the ordered pair $\tuple {\alpha, \beta}$ can be chosen from the cartesian product $S \times T$ in $m n$ different ways.
External Direct Product of Abelian Groups is Abelian Group
Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups. Then the group direct product $\struct {G \times H, \circ}$ is abelian  both $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ are abelian.
Group Direct Product of Cyclic Groups
Let $G$ and $H$ both be finite cyclic groups with orders $n = \order G$ and $m = \order H$ respectively. Then their group direct product $G \times H$ is cyclic  $g$ and $h$ are coprime, that is, $g \perp h$.
Group Direct Product of Infinite Cyclic Groups
The group direct product of two infinite cyclic groups is not cyclic.
Order of Group Element in Group Direct Product
Let $G$ and $H$ be finite groups. Let $g \in G: \order g = m, h \in H: \order h = n$. Then $\order {\tuple {g, h} }$ in $G \times H$ is $\lcm \set {m, n}$.
Subgroup Product is Internal Group Direct Product iff Surjective
Let $G$ be a group. Let $\sequence {H_n}$ be a sequence of subgroups of $G$. Let $\displaystyle \phi: \prod_{k \mathop = 1}^n H_k \to G$ be a mapping defined by: :$\displaystyle \map \phi {h_1, h_2, \ldots, h_n} = \prod_{k \mathop = 1}^n h_k$ Then $\phi$ is surjective : : $\displaystyle G = \prod_{k \mathop = 1}^n H_k$ That is,  $G$ is the internal group direct product of $H_1, H_2, \ldots, H_n$.
Internal Group Direct Product is Injective
Let $G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let $\phi: H_1 \times H_2 \to G$ be a mapping defined by: :$\map \phi {h_1, h_2} = h_1 h_2$ Then $\phi$ is injective : :$H_1 \cap H_2 = \set e$
Internal Group Direct Product Isomorphism
Let $G$ be a group. Let $H_1, H_2$ be subgroups of $G$. Let $\phi: H_1 \times H_2 \to G$ be the mapping defined by $\map \phi {h_1, h_2} := h_1 h_2$. If $\phi$ is a (group) isomorphism, then both $H_1$ and $H_2$ are normal subgroups of $G$.
Internal Group Direct Product of Normal Subgroups
Let $G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let $\phi: H_1 \times H_2 \to G$ be a mapping defined by $\map \phi {h_1, h_2} = h_1 h_2$. Let $H_1$ and $H_2$ be normal subgroups of $G$, and let $H_1 \cap H_2 = \set e$. Then $\phi$ is a (group) homomorphism.
Internal Direct Product Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $H_1, H_2 \le G$. Then $G$ is the internal group direct product of $H_1$ and $H_2$ : :$(1): \quad G = H_1 \circ H_2$ :$(2): \quad H_1 \cap H_2 = \set e$ :$(3): \quad H_1, H_2 \lhd G$ where $H_1 \lhd G$ denotes that $H_1$ is a normal subgroup of $G$.
Inclusion Mapping is Surjection iff Identity
Let $T$ be a set. Let $S\subseteq T$ be a subset. Let $i_S: S \to T$ be the inclusion mapping. Then: :$i_S: S \to T$ is surjective  $i_S: S \to T = I_S: S \to S$ where $I_S: S \to S$ denotes the identity mapping on $S$. Alternatively, this theorem can be worded as: :$i_S: S \to S = I_S: S \to S$ It follows directly that from Surjection by Restriction of Codomain, the surjective restriction of $i_S: S \to T$ to $i_S: S \to \Img {i_S}$ is itself the identity mapping.
Internal Direct Product Generated by Subgroups
Let $G$ be a group whose identity is $e$. Let $\sequence {H_n}$ be a sequence of subgroups of $G$. Then: :the subgroup generated by $\displaystyle \bigcup_{k \mathop = 1}^n H_k$ is the internal group direct product of $\sequence {H_n}$ : :$\sequence {H_n}$ is an independent sequence of subgroups such that every element of $H_i$ commutes with every element of $H_j$ whenever $1 \le i < j \le n$.
Internal Group Direct Product Commutativity
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $H_1, H_2 \le G$. Let $\struct {G, \circ}$ be the internal group direct product of $H_1$ and $H_2$. Then: :$\forall h_1 \in H_1, h_2 \in H_2: h_1 \circ h_2 = h_2 \circ h_1$
Internal and External Group Direct Products are Isomorphic
Let $G$ be a group whose identity is $e$. Then $G$ is the (external) group direct product of $G_1, G_2, \ldots, G_n$  $G$ is the internal group direct product of $N_1, N_2, \ldots, N_n$ such that: :$\forall i \in \N_n: N_i \cong G_i$ where: :$\cong$ denotes (group) isomorphism :$\N_n$ denotes $\set {1, 2, \ldots, n}$
Pullback of Quotient Group Isomorphism is Subgroup
Let $\struct {G, \circ}$ be a group whose identity element is $e_G$. Let $\struct {H, *}$ be a group whose identity element is $e_H$. Let $N \lhd G, K \lhd H$ be normal subgroups of $G$ and $H$ respectively. Let: :$G / N \cong H / K$ where: :$G / N$ denotes the quotient of $G$ by $N$ :$\cong$ denotes group isomorphism. Let $\theta: G / N \to H / K$ be such a group isomorphism. Let $G \times^\theta H$ be the pullback of $G$ and $H$ via $\theta$. Then $G \times^\theta H$ is a subgroup of $G \times H$.
Group/Examples/x+y over 1+xy
Let $G = \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\circ: G \times G \to \R$ be the binary operation defined as: :$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$ The algebraic structure $\struct {G, \circ}$ is a group.
Group/Examples/inv x = 1 - x
Let $S = \set {x \in \R: 0 < x < 1}$. Then an operation $\circ$ can be found such that $\struct {S, \circ}$ is a group such that the inverse of $x \in S$ is $1 - x$.
Group/Examples/Self-Inverse and Cancellable Elements
Let $S$ be a set with an operation which assigns to each $\tuple {a, b} \in S \times S$ an element $a \ast b \in S$ such that: : $(1): \quad \exists e \in S: a \ast b = e \iff a = b$ : $(2): \quad \forall a, b, c \in S: \paren {a \ast c} \ast \paren {b \ast c} = a \ast b$ Then $\struct {S, \circ}$ is a group, where $\circ$ is defined as $a \circ b = a \ast \paren {e \ast b}$.
Complex Numbers under Addition form Abelian Group
Let $\C$ be the set of complex numbers. The structure $\struct {\C, +}$ is an infinite abelian group.
Non-Zero Complex Numbers under Multiplication form Abelian Group
Let $\C_{\ne 0}$ be the set of complex numbers without zero, that is: :$\C_{\ne 0} = \C \setminus \set 0$ The structure $\struct {\C_{\ne 0}, \times}$ is an infinite abelian group.
Real Numbers under Addition form Abelian Group
Let $\R$ be the set of real numbers. The structure $\struct {\R, +}$ is an infinite abelian group.
Non-Zero Real Numbers under Multiplication form Abelian Group
Let $\R_{\ne 0}$ be the set of real numbers without zero: :$\R_{\ne 0} = \R \setminus \set 0$ The structure $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.
Rational Numbers under Addition form Abelian Group
Let $\Q$ be the set of rational numbers. The structure $\struct {\Q, +}$ is a countably infinite abelian group.
Non-Zero Rational Numbers under Multiplication form Abelian Group
Let $\Q_{\ne 0}$ be the set of non-zero rational numbers: :$\Q_{\ne 0} = \Q \setminus \set 0$ The structure $\struct {\Q_{\ne 0}, \times}$ is a countably infinite abelian group.
Integers under Multiplication form Countably Infinite Commutative Monoid
The set of integers under multiplication $\struct {\Z, \times}$ is a countably infinite commutative monoid.
Additive Group of Rationals is Subgroup of Reals
Let $\struct {\Q, +}$ be the additive group of rational numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Then $\struct {\Q, +}$ is a normal subgroup of $\struct {\R, +}$.
Additive Group of Integers is Subgroup of Rationals
Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\Q, +}$ be the additive group of rational numbers. Then $\struct {\Z, +}$ is a subgroup of $\struct {\Q, +}$.
Multiplicative Group of Reals is Subgroup of Complex
Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers. Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers. Then $\struct {\R_{\ne 0}, \times}$ is a normal subgroup of $\struct {\C_{\ne 0}, \times}$.
Multiplicative Group of Rationals is Subgroup of Reals
Let $\struct {\Q_{\ne 0}, \times}$ be the multiplicative group of rational numbers. Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers. Then $\struct {\Q_{\ne 0}, \times}$ is a normal subgroup of $\left({\R_{\ne 0}, \times}\right)$.
Circle Group is Infinite Abelian Group
The circle group $\struct {K, \times}$ is an uncountably infinite abelian group under the operation of complex multiplication.
Homomorphism from Reals to Circle Group
Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {K, \times}$ be the circle group. Let $\phi: \struct {\R, +} \to \struct {K, \times}$ be the mapping defined as: :$\forall x \in \R: \map \phi x = e^{i x}$ Then $\phi$ is a (group) homomorphism.
Integers under Addition form Infinite Cyclic Group
The additive group of integers $\struct {\Z, +}$ is an infinite cyclic group which is generated by the element $1 \in \Z$.
Inverse of Generator of Cyclic Group is Generator
Let $\gen g = G$ be a cyclic group. Then: :$G = \gen {g^{-1} }$ where $g^{-1}$ denotes the inverse of $g$. Thus, in general, a generator of a cyclic group is not unique.
Generators of Infinite Cyclic Group
Let $\gen g = G$ be an infinite cyclic group. Then the only other generator of $G$ is $g^{-1}$. Thus an infinite cyclic group has exactly $2$ generators.
Subgroup of Integers is Ideal
Let $\struct {\Z, +}$ be the additive group of integers. Every subgroup of $\struct {\Z, +}$ is an ideal of the ring $\struct {\Z, +, \times}$.
Additive Group of Integers is Subgroup of Reals
Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\R, +}$ be the additive group of real numbers. Then $\struct {\Z, +}$ is a subgroup of $\struct {\R, +}$.
Quotient Group of Reals by Integers is Circle Group
Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $K$ be the circle group. Then the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$ is isomorphic to $K$.
Integers Modulo m under Addition form Cyclic Group
Let $\Z_m$ be the set of integers modulo $m$. Let $+_m$ be the operation of addition modulo $m$. Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$. Then $\struct {\Z_m, +_m}$ is a cyclic group of order $m$, generated by the element $\eqclass 1 m \in \Z_m$.
Integers Modulo m under Multiplication form Commutative Monoid
The structure: :$\struct {\Z_m, \times}$ (where $\Z_m$ is the set of integers modulo $m$) is a commutative monoid.
Multiplicative Inverse in Ring of Integers Modulo m
Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$. Then $\eqclass k m \in \Z_m$ has an inverse in $\struct {\Z_m, \times_m}$  $k \perp m$.
Reduced Residue System is Subset of Set of All Residue Classes
Let $\Z_m$ be the set of set of residue classes modulo $m$. Let $\Z'_m$ be the reduced residue system modulo $m$. Then: :$\forall m \in \Z_{> 1}: \O \subset \Z'_m \subset \Z_m$
Reduced Residue System under Multiplication forms Abelian Group
Let $\Z_m$ be the set of set of residue classes modulo $m$. Let $\struct {\Z'_m, \times}$ denote the multiplicative group of reduced residues modulo $m$. Then $\struct {\Z'_m, \times}$ is an abelian group, precisely equal to the group of units of $\Z_m$.
Ring of Integers Modulo Prime is Field
Let $m \in \Z: m \ge 2$. Let $\struct {\Z_m, +, \times}$ be the ring of integers modulo $m$. Then: :$m$ is prime : :$\struct {\Z_m, +, \times}$ is a field.
Subgroups of Additive Group of Integers
Let $\struct {\Z, +}$ be the additive group of integers. Let $n \Z$ be the additive group of integer multiples of $n$. Every non-trivial subgroup of $\struct {\Z, +}$ has the form $n \Z$.
Integer Multiples under Addition form Infinite Cyclic Group
Let $n \Z$ be the set of integer multiples of $n$. Then $\struct {n \Z, +}$ is a countably infinite cyclic group. It is generated by $n$ and $-n$: :$n \Z = \gen n$ :$n \Z = \gen {-n}$ Hence $\struct {n \Z, +}$ can be justifiably referred to as the additive group of integer multiples.
Quotient Group of Integers by Multiples
Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {m \Z, +}$ be the additive group of integer multiples of $m$. Let $\struct {\Z_m, +_m}$ be the additive group of integers modulo $m$. Then the quotient group of $\struct {\Z, +}$ by $\struct {m \Z, +}$ is $\struct {\Z_m, +_m}$. Thus: :$\index \Z {m \Z} = m$
Euler's Theorem
Let $a, m \in \Z$ be coprime integers: $a \perp m$. Let $\map \phi m$ be the Euler $\phi$ function of $m$. Then: :$a^{\map \phi m} \equiv 1 \pmod m$
Symmetry Group is Group
Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\circ$ denote composition of mappings. The symmetry group $\struct {S_P, \circ}$ is indeed a group.
Area of Square
A square has an area of $L^2$ where $L$ is the length of a side of the square. Thus we have that the area is a function of the length of the side: :$\forall L \in \R_{\ge 0}: \map \Area L = L^2$ where it is noted that the domain of $L$ is the set of non-negative real numbers.
Order of Symmetric Group
Let $S$ be a finite set of cardinality $n$. Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$. Then $\struct {\map \Gamma S, \circ}$ has $n!$ elements (see factorial).
Powers of Permutation Element
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$, and let $i \in \N^*_n$. Let $k \in \Z: k > 0$ be the smallest such that: : $\pi^k \left({i}\right) \in \left\{{i, \pi \left({i}\right), \pi^2 \left({i}\right), \ldots, \pi^{k-1} \left({i}\right)}\right\}$ Then $\pi^k \left({i}\right) = i$.
Equality of Cycles
Let $S_n$ denote the symmetric group on $n$ letters, realised as the permutations of $\left\{{1, \ldots, n}\right\}$. Let $\rho = \begin{bmatrix} a_0 & \cdots & a_{k-1} \end{bmatrix}$, $\sigma = \begin{bmatrix} b_0 & \cdots & b_{k-1} \end{bmatrix} \in S_n$ be $k$-cycles of $S_n$. For any $d \in \Z$, by Integer is Congruent to Integer less than Modulus we can associate to $d$ a unique integer $\tilde d \in \left\{{0, \ldots, k-1}\right\}$ such that $d \equiv \tilde d \pmod k$. Define $a_d$ and $b_d$ for any $d \in \Z$ by $a_d = a_{\tilde d}$ and $b_d = b_{\tilde d}$ Choose $i, j \in \left\{{1, \ldots, k}\right\}$ such that: :$\displaystyle a_i = \min\left\{ {a_0, \ldots, a_{k-1} }\right\}$ :$\displaystyle b_j = \min\left\{ {b_0, \ldots, b_{k-1} }\right\}$ Then $\rho = \sigma$  for all $d \in \Z$, $a_{i + d} = b_{j + d}$. That is, $\rho = \sigma$  they are identical when written with the lowest element first.
Identity Permutation is Disjoint from All
Let $S_n$ denote the symmetric group on $n$ letters. Let $e \in S_n$ be the identity permutation on $S_n$. Then $e$ is disjoint from every permutation $\pi$ on $S_n$ (including itself).
Disjoint Permutations Commute
Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho, \sigma \in S_n$ such that $\rho$ and $\sigma$ are disjoint. Then $\rho \sigma = \sigma \rho$.
Permutation Induces Equivalence Relation
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\mathcal R_\pi$ be the relation defined by: :$i \mathrel {\mathcal R_\pi} j \iff \exists k \in \Z: \map {\pi^k} i = j$ Then $\mathcal R_\pi$ is an equivalence relation.
Existence and Uniqueness of Cycle Decomposition
Let $S_n$ denote the symmetric group on $n$ letters. Every element of $S_n$ may be uniquely expressed as a cycle decomposition, up to the order of factors.
Order of Product of Disjoint Permutations
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi$ be a product of disjoint permutations of orders $k_1, k_2, \ldots, k_r$. Then: :$\order \pi = \lcm \set {k_1, k_2, \ldots, k_r}$ where: :$\order \pi$ denotes the order of $\pi$ in $S_n$ :$\lcm$ denotes lowest common multiple.
Group Action defines Permutation Representation
Let $\map \Gamma X$ be the set of permutations on a set $X$. Let $G$ be a group. Let $\phi: G \times X \to X$ be a group action. For $g \in G$, let $\phi_g: X \to X$ be the mapping defined as: :$\map {\phi_g} x = \map \phi {g, x}$ Let $\tilde \phi: G \to \map \Gamma X$ be the mapping associated to $\phi$, defined by: :$\map {\tilde \phi} g := \phi_g$ Then $\tilde \phi$ is a group homomorphism.
Group Action determines Bijection
Let $*$ be a group action of $G$ on $X$. Then each $g \in G$ determines a bijection $\phi_g: X \to X$ given by: :$\map {\phi_g} x = g * x$ Its inverse is: :$\phi_{g^{-1} }: X \to X$. These bijection are sometimes called '''transformations''' of $X$.