Extensions of a theorem of P. Hall on indexes of maximal subgroups

Antonio Beltrán, Changguo Shao

Published 2025-01-04Version 1

We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group $G$ is a prime or the square of a prime, then $G$ is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of $G$ is nilpotent instead of having prime or squared-prime index, then $G$ continues to be solvable. Likewise, we obtain the solvability of $G$ when we assume that every proper non-maximal subgroup of $G$ lies in some subgroup of index prime or squared prime.