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.