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.