I've gotten a little tripped on part of what I'm guessing is a straight forward problem. This is part (a) of Exercise 23 in Lang's Algebra.
Let $P,P'$ be $p$-Sylow subgroups of a finite group $G$. If $P'\subset N(P)$ (normalizer of $P$), then $P'=P$.
How can I go about this? I know that $P'=gPg^{-1}$ for some $g\in G$, and I figure conjugation must come into play if the normalizer is part of the question. I also figure it'd be enough to show one is contained in the other since they're both maximal $p$-subgroups. I haven't really been able to scratch out more than that.
I'd appreciate any hints or tips/tricks or answers to this part if you have them. Thanks.