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Normal subgroup with prime index

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that if p is a prime and G is a group of order p^a for some a in Z+, then every subgroup of index p is normal in G.


    2. Relevant equations
    We know the order of H is p^(a-1). H is a maximal subgroup, if that matters.


    3. The attempt at a solution
    Suppose H≤G and (G:H)=p but H is not a normal subgroup of G. So for some g in G Hg≠gH. I know I didn't do much, but is this the correct way to start? What to do now?
     
  2. jcsd
  3. Feb 28, 2012 #2

    jbunniii

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    Do you know any general theorems about normalizers in p-groups?
     
  4. Feb 28, 2012 #3
    I am not sure about which theorem you are talking about, but I just found a theorem giving me the result I want in Dummit and Foote stating that if n is the order of the group and p the largest prime dividing n, then I have the result I wanted.
     
  5. Feb 28, 2012 #4

    jbunniii

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    The theorem I was talking about is that "normalizers grow" in p-groups. This means that if G is a p-group and H < G is a proper subgroup, then [itex]H < N_G(H)[/itex], i.e. H is a proper subgroup of its normalizer. (This is in fact true if G is any finite nilpotent group.)

    Therefore if H < G and H has index p, [itex]N_G(H)[/itex] must be all of G.
     
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