Hello I understand $A_n$ to an extend however I don't know how to prove this results and even when I have tried to verify it in $A_4$ it did not seem to commute?
Another similar proof I have struggled with is below, I think the result of one may lead to the other however I am not sure:
$1)$Prove that if a permutation $f \in A_n$ contains two independent cycles of the same length then $f$ commutes with an odd permutation
$2)$Prove that if a permutation $f \in A_n$ contains independent cycles of pairwise distinct odd lengths then $f$ does not commutes with odd permutations.
Any help would be appreciated!