The Action Of A Group On A Set

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

usually denoted by

such that for all and :

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

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

is an equalalence relation.
(ii) For each ,

is a subgroup of .

EXAMPLES. If a group acts on itself by conjugation, then the orbit of is called the conjugacy class of x. If a subgroup acts on by conjugation the stabilizer is called the centralizer of x in H and is denoted .If , is simply called the centralizer of x. If acts by conjugation on the set of all subgroups of ,then the subgroup

**normalizer of K in H** and is denoted

Sylow Theorem