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?

Given a group $G$ with $a\in G$ fixed, define $\phi: G \to G$ by $\phi(x) = axa^{-1}, x \in G$, prove that $\phi$ is always an isomorphism.

You wrote:

• 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?

That gives you that $\phi$ is bijective if $G$ is finite. Otherwise, you can prove that $\phi$ is bijective by establishing that $\phi^{-1}$ exists.

To prove that it is always an isomorphism: you also need to show that $\phi$ is a homomorphism: that is, that $\phi$ satisfies the following property: $$\phi(xy) = \phi(x)\phi(y), \forall x, y \in G.$$ The proof is fairly straightforward:

Let $x, y$ be any elements in $G$. Then:

$$\phi(x)\phi(y) = (axa^{-1})(aya^{-1}) = ax(a^{-1}a)ya^{-1} = a(xy)a^{-1} = \phi(xy)$$

== $\,\phi\,$ is $\,1-1\,$: $\,\phi(x)=axa^{-1}=1\Longrightarrow x=a^{-1}a=1\Longrightarrow \ker\phi=\{1\}\,$

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