# Math Help - Conjugation a Isomorphism?

1. ## Conjugation a Isomorphism?

Hey,

I'm having a bit of trouble w/ this one....

Let G be a group and let a be in G. Define a map Phi: G -> G by Phi(g) = a^-1 g a. Is phi an isomorphism? If so, prove it.

2. Originally Posted by bakerconspiracy
Hey,

I'm having a bit of trouble w/ this one....

Let G be a group and let a be in G. Define a map Phi: G -> G by Phi(g) = a^-1 g a. Is phi an isomorphism? If so, prove it.

What's the problem? Indeed, $\phi(gh)=\phi(g)\phi(h)\,,\,\,\phi(g)=\phi(h)\Long leftrightarrow g=h\,,\,\,and\,\,\forall\,g\in G\,\exists\,h\in G\,\,s.t.\,\,\,\phi(h)=g\Longrightarrow$ conjugation is an automorphism.