Conjugation action on itself

A group acts on itself by conjugation g·x = gxg^{-1}

Conjugation action on itself

Proposition (Conjugation action). Let $G$ be a group . Define a map $G\times G\to G$ by

g\cdot x := gxg^{-1}.

Then this defines a group action of $G$ on itself, called the conjugation action .

Context. The orbits of this action are the conjugacy classes in $G$ , and stabilizers are centralizers. This action is the mechanism behind the class equation and many counting arguments.

Proof sketch. Identity: $e\cdot x=exe^{-1}=x$ . Compatibility:

(g_1g_2)\cdot x = g_1(g_2xg_2^{-1})g_1^{-1} = g_1\cdot(g_2\cdot x).