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    prove

    prove that in any group, an element and its inverse have the same order.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by mandy123 View Post
    prove that in any group, an element and its inverse have the same order.
    let a be in G and let n\in \mathbb{Z}^+ be the order of a, i.e. n be the smallest positive integer such that a^n=e

    then (a^{-1})^n = e.

    suppose there exists a positive integer m <n such that (a^{-1})^m= e. then \underbrace{a^{-1}a^{-1}\cdots a^{-1}}_{m \mbox{ copies}} = e implies that \underbrace{aa\cdots a}_{m \mbox{ copies}} = a^m = e which is a contradiction.

    hence, order of a^{-1} is also n.
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