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Does there exist an $n$ such that there is a subgroup $G \subset S_n$ where

  1. G is non-solvable, and
  2. G contains both an odd and an even permutation?
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    Any finite group can be embedded in some $S_n$, in particular non-solvable ones can. And should the image be contained in the alternating group, take a direct product with $S_2$ which acts by interchanging two points; this cannot have made the group solvable. So it is not clear how the conditions you require make it in any way _difficult_ to embed.2012-10-16

2 Answers 2

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Taking $\,S_n\,$ as a group of permutations on the set $\,\{1,2,...,n\}\subset\Bbb N\,$ , take

$G:=\{\sigma\in S_n\;:\;\sigma(n)=n\}\cong S_{n-1}\,\,,\,\,n\geq 6$

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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 $S_{n+1}$.

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    I didn't think this way before+2013-02-15