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Math Help - Any group of order p^2 is abelian

  1. #1
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    Any group of order p^2 is abelian

    I have been asked to prove that a group of order p^2, where p is a prime, is abelian.

    This is what I have tried so far.
    G is a group of order p^2. Now Z be the center of G. So Z is a subgroup of G. So by Lagrange Theorem o(Z)|o(G).

    So o(Z) = 1, p or p^2.

    o(Z) cannot be 1 since a group whose order is a prime power has a non trivial center.

    If it is p^2 then the result is true.

    BUt how can I eliminate the possibility that o(Z) = p??
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  2. #2
    Member Black's Avatar
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    If |Z(G)|=p, then |G/Z(G)|= p \Longrightarrow G/Z(G) is cyclic \Longrightarrow G is abelian.
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