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Math Help - Group Theory

  1. #1
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    Group Theory

    Suppose a and b belong to a group, a has odd order, and aba^{-1} = b^{-1}. Show that b^2 = e.

    I have a feeling I'm missing something obvious, but I can't for the life of me figure out what a having odd order has to do with this.
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  2. #2
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    Quote Originally Posted by spoon737 View Post
    Suppose a and b belong to a group, a has odd order, and aba^{-1} = b^{-1}. Show that b^2 = e.

    I have a feeling I'm missing something obvious, but I can't for the life of me figure out what a having odd order has to do with this.
    1) aba^{-1} = b^{-1}.

    2) a^2ba^{-2} = a(aba^{-1})a^{-1} = ab^{-1} a^{-1} = (aba^{-1})^{-1} = (b^{-1})^{-1}=b.

    3) a^3ba^{-3} = a(a^2ba^{-2})a^{-1} = aba^{-1} = b^{-1}.

    4) a^4ba^{-4} = a(a^3ba^{-3})a^{-1} = ab^{-1}a^{-1} = (aba^{-1})^{-1} = b.

    The pattern is obvious: a^{2n}ba^{-2n} = b and a^{2n+1}ba^{-(2n+1)} = b^{-1}.

    Let k be order of a then b = a^k b a^{-k} = b^{-1} \implies b^2 = e.
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