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