Let $\sigma\in S_n$. For each $i\in\{1,2,\ldots,n\}$ let $k_i$ be the smallest positive integer such that $\sigma^{k_i}(i)=i$. Suppose now that $k_1,\ldots,k_n$ are all even. Is it true that $n$ must be even too?
My attempt: Suppose that $\sigma=\sigma_1\cdots \sigma_r$ is the decomposition of $\sigma$ into disjoint cycles. Can I deduce that each cycle is of even order?
