Problem 9

Let $G$ be a finite group with subgroups $H$ and $G$. Suppose the order of $H$ and the order of $K$ are coprime.

1. Show that the intersection of the subgroups $H$ and $K$ contains only the identity.

2. Show that, if $H$ and $K$ are normal subgroups of $G$, then for any $h\in H$ and $k\in K$ we have $hk=kh$.