I am trying to describe the kernel of the homomorphism $f:G \rightarrow G$ s.t. $G$ is abelian and $f(x) = x^5$.
So far I have that
$ ker(f) = \{x : x^5 = e \} = \{x : x = xe = x(x^5) = x^{6 + 5k} \} $
But this isn't making use of the abelianness of $G$ nor does it seem that interesting (which I assume the problem is getting at something interesting). Is there something I'm missing?