Why cannot the permutation $f^{-1}(1,2,3,5)f$ be even Please help me to prove that if $f\in S_6$ be arbiotrary permutation so the permutation $f^{-1}(1,2,3,5)f$ cannot be an even permutation.
I am sure there is a small thing I am missing it. Thank you.

I think, you can do the problem, if you know that:

$f$ is even so is $f^{-1}$ and $f$ is odd so is $f^{-1}$.

