Inner Automorphism Order

Question

My question is:

Does the order of the inner automorphism corresponding to some element $g$ always divide the order of $g$? If true, could you provide a proof, and if false, could you provide a counter example?

Thanks!

Answer 1

Yes.

Let $g \in G$. Then if $n$ is the order of $g$, $\alpha(h) = g h g^{-1}$, we have that $\alpha^n(h) = g^n h g^{-n} = h$.

So $\alpha^n$ is the identity.

Therefore, the order of $\alpha$ divides $n$.

Answer 2

Consider the map

$$G\to \operatorname{Inn}(G): g\mapsto (x\mapsto g^{-1}x g)$$

Check that this is a group morphism. What do you know about the order of the image of an element?