1.4 Group Action on a Set

Theorem

Let $G$ be a group. If $G$ acts on $S$, then

• The relation on $S$ defined by $x\sim x^{\prime }$ iff $gx=x^{\prime }$ for some $g\in G$ is an equivalent relationship.
• For $x\in S$, let $G_x=\{h\in G:hx=x\}$. Then $G_x$ is a subgroup of $G$.

Definition

Denote the equivalence class of $x\in S$ by $\overline{x}=\{gx:g\in G\}$, and it is called the orbit of $x$.

Examples. Let $G$ be a group and $H$ be a subgroup of $G$.

• $H$ acts on $G$ by conjugation. Then $H_x:=\{h\in H:hx{h}^{-1}=x\}$ is the centralizer of $x$ in $H$.
• $G$ acts on $\{\text{mbox}allsubgroupsofG\}$ by conjugation. Then $G_K:=\{g\in G:K^g=K\}$ is the normalizer of $K$.

Theorem

Let $G$ act on $S$. Then $\#|\overline{x}|=[G:G_x]$.

\begin{proof} Define a map $\{\text{mbox}allleftcosetsof{G}_x\}\to \overline{x}$ by $g{G}_x\mapsto gx$. It is easy to show that this map is bijective. \end{proof}

Corollary

Let $G$ be a finite group. Consider $G$ acting on $G$ by conjugation.

• number of elements in conjugate class of $x\in G$ is $[G:C_G(x)]$.
• Suppose $\overline{x_1},\cdots ,\overline{x_n}$ are all distinct conjugacy classes of $G$. Then $|G|={\sum }_{i=1}^n[G:C_G(x_i)]$.
• Let $K<G$. Then the number of subgroup of $G$ conjugate to $K$ is $[G:N_G(K)]$,

Theorem

$G$ acting on $S$ induces a group homomorphism $G\to \mathcal{A}(S)$, where $\mathcal{A}(S)$ is a group of all permutations of $S$.

Cayley

Let $G$ be a group. Then there exists a group homomorphism $G\hookrightarrow \mathcal{A}(G)$. In particular, if $|G|=n$, then $G\hookrightarrow S_n$,

Notation. $\mathcal{A}(G)=\mathrm{Perm}(G)$.

Remark.

• $\mathrm{Inn}(G)⊲\mathrm{Aut}(G)$ and $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)$
• Class equation can be rewritten as $|G|=|C_G(G)|+{\sum }_{x\notin C(G)}[G:C_G(x)]$.