Does there exist an $n$ such that there is a subgroup $G \subset S_n$ where
- G is non-solvable, and
- G contains both an odd and an even permutation?
Does there exist an $n$ such that there is a subgroup $G \subset S_n$ where
Taking $\,S_n\,$ as a group of permutations on the set $\,\{1,2,...,n\}\subset\Bbb N\,$ , take
$G:=\{\sigma\in S_n\;:\;\sigma(n)=n\}\cong S_{n-1}\,\,,\,\,n\geq 6$
Take any nonsolvable group $G$ of order $n=2^am$, where $a>0$ and $m>1$ is odd. By Cayley's theorem, $G\leqslant S_n < S_{n+1}.$ $G$ contains both even and odd permutations by Cauchy's theorem and is a proper nonsolvable subgroup of $S_{n+1}$.