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By Cayley's embedding theorem every group $G$ embeds into the symmetric group $S_{|G|}$. But which groups $G$ have the property that there exists some $n$ such that $G$ is a quotient of $S_n$? My intuition is that it couldn't possibly be all finite groups. Is there a nice characterization of the groups that do have this property?

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You should probably ask this on math.stackexchange.com Also, reading about the alternating groups and their simplicity will be of help! –  Mariano Suárez-Alvarez Mar 12 '12 at 19:10
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closed as too localized by Bruce Westbury, Mariano Suárez-Alvarez, Neil Strickland, Qiaochu Yuan, Tom Leinster Mar 12 '12 at 19:34

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The symmetric groups $S_n$ for $n \leq 4$ are solvable. But for $n >4,$ the only proper non-trivial normal subgroup of $S_n$ is the alternating group $A_n.$ Hence the only factor groups of symmetric groups, other than $S_n$ itself and the trivial group are: the cyclic group of order $2$ (a non-trivial factor group of every $S_n$ with $n >2)$, and the group $S_3$ (which occurs in a non-trivial way as a factor group of $S_4).$ These elementary facts can be found (at least implicitly) in almost any algebra text.

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