question about $p$-Sylow subgroups

Question

I was wondering whether it is necessarily true that if $P_1$ and $P_2$ are Sylow $p$-subgroups of $G$ then $P_2\subseteq N_G(P_1)$. I don't think it is because since they're both Sylow $p$-groups, they can't be normal. I think that $P_2$ would not be in $N_G(P_1)$ then. Some help? Thanks!

Answer 1

There's the following general fact about $p$-groups and $p$-sylow groups.

If $G$ is a group, $H \leq G$ is a $p$-Sylow subgroup, for some prime $p$, then for every $K \leq G$a $p$-subgroup such that $K \leq N_G(H)$ then $K \leq H$.

Exercise: prove it. :)