If a group has order $2p^a$ where p is an odd prime and a $\geq$ 1, then G has a proper, nontrivial normal subgroup

Question

I have this problem and I'm stuck. I only know that $Z(G)$ is a normal subgroup of $G$ but don't know how to continue from there.

I would really appreciate any help.

Answer 1

The Sylow $p$-group has index 2, so it has to be normal.