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Some more p-Sylow subgroups and Normalizers

  1. Dec 5, 2008 #1
    1. The problem statement, all variables and given/known data

    Let G be a finite group, H a normal subgroup of G and B a p-Sylow subgroup of H for some p dividing |H|.

    1. Let q be a prime dividing |G|, q != p, Q be a q-Sylow subgroup of G. Prove that Q acts on B by conjugation.
    2. Let Q' be the image of Q in G/H, show that if |Q'|=|Q| then KxB is (isomorphic to) a subgroup of G, where K is a kernel of the action in (1).
    3. Suppose Q is normal in G, does that imply that the image of KxB in G/H is normal in G/H?


    2. Relevant equations

    The standard theory of the Sylow theorems.
    In the previous parts of the question morphism here helped me prove that G=HN_G(B) where N_G(B) is the normalizer of B in G, and that G/H is isomorphic to N_G(B)/N_H(B)

    3. The attempt at a solution

    (1) Actually, I think there must be some mistake in the question. For Q to act on B we need that Q be a subset of the normalizer of B in G, and that doesn't seem to be the case. It might be that I'm missing something here...

    (2) This would imply that the intersection of Q and H is trivial, but I'm not sure how K would "look like" to gain intuition on how this could unfold.

    (3) I think not, but if I could present G as some direct product that might give me a reason why.
     
  2. jcsd
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