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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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There is a cyclic group of order 6 in $S_5$. –  S. Carnahan Jan 17 '12 at 9:03
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Is it actually known how to compute the least $n$ such that $G$ is a subgroup of $S_n?$ –  Igor Rivin Jan 17 '12 at 12:53
    
@Igor: I think, its not completely known (this problem (your) was posed on mathoverflow). In group theory, if we want to show that a group is isomorphic to $A_n$ or $S_n$, it follows easily if we have subgroup of index $n$. But "Is it correct step to show existence of index n subgroup?", this was a question I wondering. –  RDK Jan 18 '12 at 5:11
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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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1 Answer

This is not true. The group $S_3 \times S_2$ can be embedded in $S_5$, but not in smaller symmetric groups.

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