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