Subgroups of $S_n$, contained in $A_n$ or half of elements are odd permutations

Question

good morning. I am unable to resolve this issue, could someone help me, please?

Question: If $H$ is a non-trivial subgroup of $S_{n}$, then or $H \subset A_{n}$ or exacly half of the elements are odd permutations.

Answer 1

We can still reason like this. If $A=A_n\cap H$ and $x\in H\setminus A_n$, then $H=A\cup xA$ and $|A|=|xA|$. Why does $H=A\cup xA$? If $h\in H$ and $h\notin A$, then $x^{-1}h\in A$. So $h\in xA$.