The Action Of A Group On A Set

Definition 4.1
An action of a group G on a set is a function

G\times S\to S

usually denoted by

(g,x)\mapsto gx

such that for all x \in Sandg{1}g{2}\in G :

ex=x \qquad\text{and} \qquad (g_1g_2)x=g_{1}(g_{2x})

When such an action is given, we say that G acts on the Set S .

Theorem 4.2. Let G be a group that acts on a set S .
(i) The relation on S defined by

x\sim x'\Longleftrightarrow gx=x'\quad\text{for some}\quad\\g\in G

is an equalalence relation.
(ii) For each x\in S ,

G_x=\{g\in G\mid gx=x\}

is a subgroup of G .

EXAMPLES. If a group G acts on itself by conjugation, then the orbit {gxg^{-1}\mid g \in G} of x \in G is called the conjugacy class of x. If a subgroup H acts on G by conjugation the stabilizer H_x={h\in H\mid hxh^{-1}=x}={h\in H\mid hx=xh} is called the centralizer of x in H and is denoted C_H(x) .If H=G ,C_G(x) is simply called the centralizer of x. If H acts by conjugation on the set S of all subgroups of G ,then the subgroup

K\in S {h\in H\mid hKh^{-1}=K} normalizer of K in H and is denoted N_H(K)

Sylow Theorem