Let G be a group, P be an abelian Sylow-p subgroup of G. Let $N=N_G(P)$ and assume that H is a complement of P in N which I believe means that $HP=N$ and $H\cap{P}=1$
Prove that $P = P_1 \times P_2$ where $P_1=C_P(H)$ and $P_2=[P,H]$
I understand that there's three things that I need to prove:
$P_1P_2=P$
$P_1\cap{P_2}=1$
elements of $P_1$ commute with elements of $P_2$
But I cannot seem to prove either of them.
I would appreciate any advice on how to approach this. Thank you.