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Group theory 2 problems

  1. Jul 24, 2010 #1
    Hi, I've been vanquished by probably easy problems once again.

    1. The problem statement, all variables and given/known data
    1. Let G be a group of order p^2 (p prime number), and H its subgroup of order p. Show that H is normal. Prove G must be abelian.

    2. If a group G has exactly one subgroup H of order k, prove H is normal in G.

    2. Relevant equations
    Lagrange theorem I think. Isomorphism theorems maybe?

    3. The attempt at a solution
    1. Obviously H is cyclic. If H is not normal, G cannot be abelian, hence all the elements are of order p, except for the neutral one. G is therefore divided into p+1 cyclic, disjoint (except for e) subgroups of order p. So far I haven't succeeded deriving a contradiction.

    2. Normalizer is either H or the whole group. Perhaps some property of self-normalizing groups yields a contradiction?

    Thank you very much for any hints.
  2. jcsd
  3. Jul 25, 2010 #2


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    The second one isn't hard at all. Just think about the conjugate subgroups of H. The first one is a little harder. I'd start by using the class equation to show that G has a nontrivial center.
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