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Conjugates in the normalizer of a p-Sylow subgroup

  1. Mar 19, 2008 #1
    [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.
    Last edited: Mar 19, 2008
  2. jcsd
  3. Mar 19, 2008 #2
  4. Mar 21, 2008 #3
    Well, let me put up some more of what I have found - still no solution.

    Since a,b [itex] \in Z(P) [/itex] we know that [itex] P \subseteq C(a) [/itex] and [itex] P \subseteq C(b)[/itex]. Since P [itex] \subseteq [/itex] N(P) we also know that [itex] P \subseteq N(P) \cap C(a) [/itex] and [itex] P \subseteq N(P) \cap C(b) [/itex].

    Further, [itex]x^{-1} C(a) x [/itex] = [itex] C(b) [/itex] so that the conjugate Sylow subgroup [itex] Q = x^{-1}Px \subseteq C(b)[/itex].

    We know that N(P) and N(Q) each have unique Sylow p-subgroups P and Q respectively. If we know in addition that C(a) and C(b) also have unique p-Sylow subgroups then this would solve the problem. However I know of no reason why this must be the case. Anybody see anything obvious that I'm missing (or not obvious :confused:)?

    Thanks again all.
  5. Mar 23, 2008 #4
    OK solved, if anyone wants to see solution let me know.
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