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Math Help - Prove G is abelian iff...

  1. #1
    Member elninio's Avatar
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    Prove G is abelian iff...

    Let G be a group. Prove that G is abelian if and only if (ab)^-1 = a^-1 * b^-1 for all a,b in G.
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  2. #2
    Super Member Gamma's Avatar
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    Quote Originally Posted by elninio View Post
    Let G be a group. Prove that G is abelian if and only if (ab)^-1 = a^-1 * b^-1 for all a,b in G.
    First note we always have (ab)^{-1}=b^{-1}a^{-1} See why?

    so if G is abelian, the forward direction is clear (just move them past eachother). Conversely, if we have a^{-1}b^{-1}=(ab)^{-1}=b^{-1}a^{-1} for all a and b in G.

    Just multiply on the left by (ba) and on the right by (ab)
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