Showing that Conjugation is a Group Action

Question

I've been pretty stuck on trying to show how a conjugation is a group action:

A group $G$ acts on itself via conjugation, where $\phi_g(x)$ = $gxg^{-1}$. Prove that conjugation is a group action.

How do I show that conjugation is a group action? Thank you.

Answer 1

You have to check two things.

• If you denote $e$ the neutral element of $G$, for all $x\in G$ you have

$$\phi_e(x)=exe^{-1}=x.$$

• And if $g,h\in G$, for all $x\in G$ you have

$$\phi_g(\phi_h(x))=\phi_g(hxh^{-1})=g(hxh^{-1})g^{-1}=(gh)x(gh)^{-1}=\phi_{gh}(x).$$

Then you have proven that it is a group action.