Cycle types of elements of centralizers of transitive permutation groups

Question

I think the following is true, but I'm not sure how to prove it. Any help would be greatly appreciated.

Suppose $G$ is a transitive subgroup of $S_n$ and $C$ is the centralizer of $G$ in $S_n$. If $\sigma\in C$ has order $k$, then $\sigma$ is a product of $n/k$ disjoint $k$ cycles.

Answer 1

This is true and well known. The property that $C$ has in your conclusion is usually just called "semiregular".

(See https://en.wikipedia.org/wiki/Group_action_(mathematics) or https://groupprops.subwiki.org/wiki/Semiregular_group_action)

So what you want is that "the centraliser of a transitive group is semiregular". Googling this phrase gives a bunch of hits.

For example, see Theorem 4.2A in the "Permutation Groups" book by Dixon and Mortimer.