This is problem 26 from Grove's "Algebra."
Suppose $K$ is a conjugacy class in $S_n$ of cycle type $(k_1,...,k_n)$, and that $K \subseteq A_n$. If $\sigma \in K$ write $L$ for the conjugacy class of $\sigma$ in $A_n$.
If either $k_{2m} > 0$ or $k_{2m+1} > 1$ for some $m$ show that $L = K$.
I can show $L \subseteq K$ but not $K \subseteq L$. I don't know how to use the "$k_{2m} > 0$ or $k_{2m+1} > 1$" hypotheses. If $k_{2m} > 0$ for some $m$ then $\sigma \in A_n$ must have an even number of odd transpositions. Can I get a hint?
Thank you.
Edit: $k_m$ is the number of cycles of length m.