If $G$ is a finite group, and $n$ is the least positive integer such that $G$ can be embedded in symmetric group $S_n$, then, should $G$ necessarily contain a subgroup of index $n$?
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closed as too localized by Mark Sapir, Ryan Budney, Martin Brandenburg, Bill Johnson, Benjamin Steinberg Jan 17 '12 at 13:09
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This is not true. The group $S_3 \times S_2$ can be embedded in $S_5$, but not in smaller symmetric groups. 

