Does there exist a finite perfect group with a subgroup of index at most 4?
Does there exist a perfect group with a subgroup of index at most 4?
0
$\begingroup$
group-theory
finite-groups
-
0Yes, $A_5$ has a subgroup of index 1. – 2012-10-21
1 Answers
2
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 $\,T'\leq N\,$