Let $H<\mathfrak S_n$ and $\sigma\in H $. Prove/disprove that every disjoint cycle in its decomposition belong to $H.$

Question

Let $H$ be a proper subgroup of $\mathfrak S_n$ and $ \sigma \in H $ such that $ \sigma = c_1c_2...c_s$ where $c_1,c_2,...,c_s$ are disjoint permutations.

I want to prove/disprove that $c_1,...,c_s \in H.$

Any idea?

Answer 1

You can't, since it is not true. Suppose that $n=4$ and that$$H=\bigl\{e,\overbrace{(1\ \ 2)(3\ \ 4)}^\sigma\bigr\}.$$Then $H$ is a proper subgroup of $S_4$, $\sigma=(1\ \ 2)(3\ \ 4)\in S$, but $(1\ \ 2),(3\ \ 4)\notin H$.