Let $S_n$, $A_n$ denote the symmetric group and alternating group, each on $n \in \mathbb{N}$ letters respectively. Is there a way to characterize all elements $\alpha \in S_n$ satisfying the equation $\{g\alpha g^{-1}|\;g\in S_n\}=\{g\alpha g^{-1}|\;g\in A_n\}\:$?
I was wondering for what elements of the symmetric group on $n$ letters one can arrive at all conjugate (in $S_n$) by conjugating only by even permutations. I know of a sufficient condition on $\alpha \in S_n$ for the above equation to be satisfied – $\alpha$ commutes with at least one odd permutation. I tried computing explicitly for a few elements in $S_3$ and $S_4$ without much fruition. Any help is appreciated. Thank you.
Best Answer
Your sufficient condition, that $a$ commutes with at least one odd permutation, is also necessary, provided $n\geq2$ so that there is an odd permutation.
To prove it, let $h$ be an odd permutation. Then, by assumption, $hah^{-1}=gag^{-1}$ for some even permutation $g$. Multiplying on the left by $h^{-1}$ and on the right by $g$, you get $ah^{-1}g=h^{-1}ga$. Thus, $a$ commutes with the odd permutation $h^{-1}g$.