The question is as follows: prove or disapprove: every group of order $135$ must be abelian.
I started like this: $G = H \times\ K$ when $H$ is a normal 5-sylow subgroup and $K$ is a normal 3-sylow subgroup (both normal from sylow theory).
$H$ is cyclic and therefor abelian, but what about $K$? if it's cyclic then it proves the statement. if not, I'm not sure how to continue...
I'm not sure how to continue from here.