I have been told to use the following to prove another claim, but I would like to prove this anyway for myself. However I can't tell why it's true. I think it's true, but can't see why! Here it is:
If $P$ and $Q$ are distinct $p$-Sylow subgroups in a group $G$ then $Q\not\subseteq N_G(P)$.
Could I use the theorem below? If so, how? I mean $Q$ isn't a $p$-subgroup, which is why I'm unsure.
If $G$ is a group, $P \leq G$ is a $p$-Sylow subgroup, for some prime $p$, then for each $H \leq G$ a $p$-subgroup such that $H \leq N_G(P)$ then $H \leq P$.