Groups with normal restriction property

TL;DR

This paper proves that if all maximal subgroups of a finite group are NR-subgroups, then the group is solvable, confirming a conjecture and utilizing the Classification of Finite Simple Groups.

Contribution

It establishes a new criterion for solvability based on the normal restriction property of maximal subgroups, resolving a conjecture from 1998.

Findings

If every maximal subgroup is an NR-subgroup, then G is solvable.

The result relies on the Classification of Finite Simple Groups.

Confirms a conjecture by Berkovich (1998).

Abstract

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every maximal subgroup of G is an NR -subgroup then G is solvable. This gives a positive answer to a conjecture posed in Berkovich (Houston J Math 24:631-638, 1998).