Is the conjugation map always an isomorphism? Given a group $G$ with $a\in G$ fixed, define $\phi: G \to G$ by $\phi(x) = axa^{-1},  x \in G$, I am wondering when $\phi$ is an isomorphism. I think that it is always an isomorphism because it is one to one, and the order of the domain ($|G|$) and the order of the range are equal. Am I wrong in thinking this?

== $\,\phi\,$ is $\,1-1\,$: $\,\phi(x)=axa^{-1}=1\Longrightarrow x=a^{-1}a=1\Longrightarrow \ker\phi=\{1\}\,$
== $\,\phi\,$ is onto: for all $\,y\in G\,\,,\,\phi(a^{-1}ya)=a(a^{-1}ya)a^{-1}=y\,$
