On the residual of a finite group with semi-subnormal subgroups

Abstract

A subgroup A of a group G is called seminormal in G, if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. We introduce the new concept that unites subnormality and seminormality. A subgroup A of a group G is called semi-subnormal in G, if A is subnormal in G or seminormal in G. In this paper, the F-residual of a group G = AB with semi-subnormal subgroups A and B such that A;B 2 F, where F is a saturated formation and U F, is studied. Here U is the class of all supersoluble groups and the F-residual of G is the intersection of all those normal subgroups N of G for which G=N 2 F.