Prove that if $K$ is a normal subgroup of $G, a ∈ G, n ∈ N,$ and $a^{n} ∈ K$. Then $|aK|$|$n$.

Question

Prove that if $K$ is a normal subgroup of $G, a ∈ G, n ∈ N,$ and $a^{n} ∈ K$. Then $|aK|$|$n$.

So I know I'm supposed to use the fact that for a group $A, h ∈ A,$ and a positive m, if $h^{m}=e$, then $|h|$|$ m$, but I'm not sure how to start it. Any help is appreciated.

Answer 1

The key idea is that $\phi:G\to G$ defined by $\phi(x)=ax$ is a bijection. What does this tell you about $|K|$ and $|aK|$?