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Normal Subgroups Proof with defined order

  1. Apr 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Let G be a group and H normal with G. Prove that if |H| = 2, then H is a subgroup of Z(G).

    2. Relevant equations



    3. The attempt at a solution

    Since H is a subgroup, it must contain the identity, call it e. Call the non-identity element of H h. Thus, H = {h, e}. Since H contains its own inverses, h^2 = e (if h^2 = h, then h would have to be the identity).

    Anyway, by the normality of H, we know that for any element g of G, gH = Hg. That is to say,
    {gh, ge} = {hg, eg}, which means that
    {gh, g} = {hg, g}

    since h is not the identity, we know that gh≠g and hg≠g. Thus, in order for Hg and gH to be equal, we must have that gh = hg for an arbitrary element g. Thus, we have shown that H is a subgroup of Z(G).

    a.) Is this right?
    b.) If someone could clean up any loose ends, that would be great. My teacher is extremely picky on homework.
     
    Last edited: Apr 18, 2012
  2. jcsd
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