This is Exercise 7, page 40 from Hungerford's book Algebra.
Let $G$ be a group of order $p^{k}m,$ with $p$ prime and $(p,m)=1.$ Let $H$ be a subgroup of order $p^{k}$ and $K$ a subgroup of order $p^{d}$, with $0
and $K\nsubseteq H.$ Show that $HK$ is not a subgroup of $G$.
I've started assuming that $HK$ is a subgroup, but it din't help me.
Thanks in advance!