I am trying to prove a theorem for my Algebra Homework and I am stuck in a specific problem, I would be much grateful if someone can help me.
If $\sigma \in A_n$ and the cycle type of $\sigma$ consists of cycles with distinct odd lenght, how can I prove (without using Group Actions) that the set of permutations that commutes with $\sigma$ is the group generated by the cycles in the cycle decomposition of $\sigma?$ Or, at least that the centralizer $C_{S_n}(\sigma)$ only admits even permutations.
The cycle decomposition in this problem includes the 1-cycles.
Thank you.