Proposition 1. Suppose that $G$ acts on $X$ through an action $\star$. Let $g \in G$. Define a map $f_g : X \to X$ by
$$
f_g(x) = g \star x .
$$
Then $f_g \in \operatorname{Sym}(X)$.

Definition 1. Let $\sigma\in S_n$, and write $\sigma$ as a product of disjoint cycles. We say that the form of $\sigma$ is
$$
1^{\lambda_1}2^{\lambda_2}\cdots n^{\lambda_n}
$$
if $\sigma$ has exactly $\lambda_r$ cycles of length $r$ for each $1\le r\le n$.

Example 1. The form of the permutation
$$
\sigma=(1\ 2\ 3)(4\ 5)
$$
in $S_7$ is
$$
1^22^13^14^05^06^07^0=1^22^13^1.
$$

Proposition 1. Let $N$ be a normal subgroup of a group $G$. Then $N$ is a maximal normal subgroup of $G$ if and only if $G/N$ is simple.

Lemma. Let $a$ be an element of largest order in a finite abelian group $G$, that is,
$$
o(a)=\max\{o(g)\mid g\in G\}.
$$
Then for every $g\in G$, we have
$$
o(g)\mid o(a),
$$
where $o(a)$ denotes the order of $a$. It follows that $o(a)$ is the least common multiple of the orders of all elements of $G$.

Proposition 1. If $G$ is an infinite cyclic group, then $\operatorname{Aut}(G)$ is a cyclic group of order $2$.

Proposition 1. Let $G$ be a group, and let $H$ be a normal subgroup of $G$. If the index $[G:H]=n<\infty$, then for every $x\in G$, we have $x^n\in H$.

Method 1:

Let $H$ be a subgroup of group $G$. Then $H \lhd G \iff \forall g \in G, \ gH = Hg \ (\text{or} \ gHg^{-1} = H)$.

Method 2:

Let $H$ be a subgroup of group $G$. Then $H \lhd G \iff \forall g \in G, h \in H, \ ghg^{-1} \in H$.

Definition 1. Let $H$ be a subgroup of a group $G$, and suppose that the index of $H$ in $G$ is $r$, that is, $[G:H]=r$. Then
$$
G=a_0H\cup a_1H\cup\cdots\cup a_{r-1}H,
$$
where $a_0=e$ and
$$
a_iH\cap a_jH=\varnothing \quad \text{for } i\ne j,
$$
is called a coset decomposition of $G$ with respect to its subgroup $H$.

Proposition 1

Let $G$ be a group, let $a\in G$, and suppose that the order of $a$ is $m$. Prove that:

(1) $a^n=e$ if and only if $m\mid n$.

(2) $a^k=a^h$ if and only if $k\equiv h \pmod m$.

Criterion 1: The Definition of a Group

Let $G$ be a nonempty set, and suppose a binary operation “$\cdot$” is defined on $G$. If this binary operation satisfies the following conditions:

(1) Closure: $ \forall x,y \in G,\ x\cdot y\in G $

(2) Associativity: $ \forall x,y,z\in G,\ (x\cdot y)\cdot z=x\cdot (y\cdot z) $

(3) Identity element: $ \exists e\in G,\ \forall a\in G,\ e\cdot a=a\cdot e=a $

(4) Inverse element: $ \forall a\in G,\ \exists b\in G,\ a\cdot b=b\cdot a=e $

then $G$ is called a group with respect to “$\cdot$”.