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Frattini subgroup

  1. Oct 9, 2009 #1
    1. The problem statement, all variables and given/known data

    G is a finite p-group, show that [tex]G/ \Phi (G)[/tex] is elementary abelian p-group.

    2. Relevant equations

    [tex]\Phi (G)[/tex] is the intersection of all maximal subgroups of G.

    3. The attempt at a solution

    By sylow's theorem's we have 1 Sylow p-subgroup which is normal, call P. Then the order of G/P = p, so it's cyclic and thus abelian. Since there is only one maximal subgroup then [tex]\Phi (G) = P[/tex]???.

    I'm having trouble convincing myself this is complete. I'm told that [tex]G/\Phi (G)[/tex] can be of prime powered order. I'm afraid i might be missing something.
  2. jcsd
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