Groups with normal restriction property

摘要

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever Ktrianglelefteq M{Ktrianglelefteq M}, then K G ∩ 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 (1998), 631–638).