Prove that f and g^-1fg, for any f,g in Sn, are of the same parity.

Please show steps. Thanks!

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- Apr 24th 2009, 07:58 AMmpryalgroup permutations 3
Prove that f and g^-1fg, for any f,g in Sn, are of the same parity.

Please show steps. Thanks! - Apr 24th 2009, 08:23 AMThePerfectHacker
Notice $\displaystyle g^{-1}(1,2)g = (g^{-1}(1),g^{-1}(2))$.

Write $\displaystyle f=t_1...t_m$ for transpositions $\displaystyle t$. Then $\displaystyle g^{-1}fg = (g^{-1}t_1g)(g^{-1}t_2g)...(g^{-1}t_mg)$. Each $\displaystyle g^{-1}t_jg$ is a transposition by the above sentence. Thus, $\displaystyle f,g^{-1}fg$ both have $\displaystyle m$ components in its transpotitions so they have the same parity.