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Does there exist a finite perfect group with a subgroup of index at most 4?

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    Yes, $A_5$ has a subgroup of index 1.2012-10-21

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Hints:

1) $\,G\,$ is perfect iff $\,G':=[G,G]=G\,$

2) For any group $\,K\,$ , if it has a subgroup $\,H\,$ of index n then it has a normal subgroup of index dividing $\,n!\,$ and, in fact, this normal subgroup is contained in $\,H\,$

3) Every group of order less than $\,5\,$ is abelian

4) For any group $\,T\,$ , if $\,N\lhd T\,$ then the quotient $\,T/N\,$ is abelian iff $\,T'\leq N\,$