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**[SOLVED] Conjugates in the normalizer of a p-Sylow subgroup**

**1. Homework Statement**

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. Homework 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.

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