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Math Help - Prove both even or both odd

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    Prove both even or both odd

    Let alpha and beta belong to S_n. Prove that (beta)(alpha)(beta^-1) and alpha are both even or both odd.
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    Quote Originally Posted by mandy123 View Post
    Let alpha and beta belong to S_n. Prove that (beta)(alpha)(beta^-1) and alpha are both even or both odd.
    Notice that \beta (a_1,a_2) \beta^{-1} = (\beta(a_1), \beta(a_2)).
    Therefore if \alpha = \tau_1 \cdot ... \cdot \tau_k where \tau_i are transpositions then \beta \alpha \beta^{-1} = (\beta \tau_1 \beta^{-1})(\beta \tau_2 \beta^{-1}) ... (\beta \tau_k \beta^{-1}). Thus, we see that \alpha and \beta \alpha \beta^{-1} have the same parity.
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