On complemented subgroups of finite groups

Abstract

In this paper, it is proved that the class of all finite supersoluble groups with elementary abelian Sylow subgroups is just the class of all finite groups for which every minimal subgroup is complemented. The structure of a finite group under the assumption that all maximal subgroups (respectively 2-maximal) of any Sylow subgroup are complemented is also analyzed.

C-Supplemented subgroups of finite groups

“…It is clear from these results that complementation of some families of subgroups of a group has a strong in¯uence on its structure. This idea was strengthened in [1], where the complementation of minimal subgroups and maximal subgroups of the Sylow subgroups is studied. The main goal of the present paper is to study the csupplemented subgroups.…”

On solvability of finite groups with some ss-supplemented subgroups

“…In fact, they proved that a group G is solvable if the Sylow 2-subgroups and Sylow 3-subgroups of G are complemented in G. Moreover, Hall in [10] proved that a group G is supersolvable with elementary abelian Sylow subgroups if and only if every subgroup of G is complemented in G. Ballester-Bolinches and Guo in [4] analysed the class of groups for which every subgroup of prime order is complemented. In fact, they proved that G is supersolvable if every subgroup of prime order of G is complemented in G.…”

A note on infinite aS-groups

“…Both aS-and aC-groups have been investigated by Kappe and Kirtland in [8]. Since the study of supplementation (complementation) in a group can reveal important properties about the structure of the group, this field has received a good deal of attention from several authors (for example see [3], [6] and [8]). In this paper, we investigate infinite aS-groups.…”