There exists a group element $\sigma\in S_7$ under some conditions?

Question

It is given that $\sigma\in S_7$ where $S_7$ is a symmetric group. Do there exist an element $\sigma$ such that $\sigma^{20}=\sigma$ and $\sigma\ne e$ where $e$ is an identity element?

My attempt: Since $\sigma^{20}=\sigma$

So, $\sigma^{19}\sigma=e\sigma$, then $\sigma^{19}=e$ because $S_7$ is a group so there exists an inverse of $\sigma$.

It follows that the order of $\sigma=19$ since $19$ is a prime number.

So $\sigma$ must be a product of disjoint cycle, since there does not exist any disjoint cycle lengths of $\sigma$ such that their L.C.M is equals to 19.

Hence, no element exist under the above condition.

Does it correct? Could anyone help me? Thanks.

Answer 1

No such element by Lagrange's theorem. The order would be $19$, but $19\not|7!$.

Answer 2

Yes, it is correct. Your logic seems right.