[SOLVED] Conjugates in the normalizer of a p-Sylow subgroup 1. The problem statement, all variables and given/known data Let P be a p-Sylow subgroup of G and suppose that a,b lie in Z(P), the center of P, and that a, b are conjugate in G. Prove that they are conjugate in N(P), the normalizer of P (also called stablilizer in other texts I believe). This is problem 29 from section 2.11 of Herstein's Abstract Algebra, the last problem in Chapter 2. 2. Relevant equations Any of the standard results in Sylow theory are usable. 3. The attempt at a solution I believe that it would be sufficient to demonstrate that if b = x^(-1)ax then x must lie in the normalizer of P, N(P). We know that if x is in N(P) then x^(-1)Px = P. We also know that both a and b commute with every element of P. In fact, we also know that x can't be in P, unless a = b. This doesn't seem like it should be too difficult, but I would love a small hint of what direction to take. Thanks all.