Prime order of a cycle in the alternating group

Question

Let $H$ be any nontrivial subgroup of the alternating group $\mathfrak{A}_n, \: n \geq 5$.

Is it true that $H$ contains an element $\sigma$ whose order is a prime $p$, that is an element of the form:

\begin{equation} \displaystyle \sigma=c_1c_2...c_r \end{equation}

where $c_i$ are disjoint $p$-cycles and $o(\sigma)=\text{lcm}\:(l_1,l_2,...,l_r)=p$ ?

Note: this is a part of a demonstration that leads to the conclusion: $H=\mathfrak{A}_n$.

Answer 1

If $p\mid n!/2$, then by Cauchy's theorem there is an element of order $p$.

Or, if $2\lt p\le n$, you can just take any $p$-cycle.