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Math Help - Group order prime, center, abelian

  1. #1
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    Group order prime, center, abelian

    Hello, I need help with this exercise

    ----- If #(G) = p^3, with p prime, then G is abelian or #(Z(G)) = p -----

    THANKS!
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  2. #2
    Super Member Matt Westwood's Avatar
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    This one comes up again and again. Try this page:

    Symmetric Group Center Trivial - ProofWiki

    SORRY - ignore me! This is a completely different problem! Silly me.
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  3. #3
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    Quote Originally Posted by roporte View Post
    Hello, I need help with this exercise

    ----- If #(G) = p^3, with p prime, then G is abelian or #(Z(G)) = p -----

    THANKS!
    Burnisde's lemma says that \text{Z}(G) = p,p^2,p^3. If it is p^3 proof complete. Thus you need to show \text{Z}(G) = p^2 is impossible.
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