On $NR^*$-subgroups of finite groups

Abstract

Let G be a finite group and let H be a subgroup of G. H is said to be an NR*-subgroup of G if there exists a normal subgroup T of G such that G = HT and if whenever K \lhd H and g \in G, then Kg \cap H \cap T\leq K. A number of new characterizations of a group G are given, under the assumption that all Sylow subgroups of certain subgroups of G are NR*-subgroups.