Finite Groups with Some -Supplemented Subgroups

Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime We say that is -supplemented in if there is a subgroup of such that and where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in . 1. Introduction Throughout only finite groups are considered. Let stand for the set of all prime divisors of the order of a group . denotes the class of all supersolvable groups. char means is a characteristic subgroup of . We use conventional notions and notation, as in Robinson [1]. Let be a class of groups. is called a formation provided that (1) if and , then and (2) if and are in , then is in for all normal subgroups , of . A formation is said to be saturated if implies that . Two subgroups and of a group are said to be permutable if . is said to be -quasinormal in if permutes with every Sylow subgroup of , that is, for any Sylow subgroup of . This concept was introduced by Kegel in [2] and has been studied widely by many authors, such as [3, 4]. An interesting question in the theory of finite groups is to determine the influence of the embedding properties of members of some distinguished families of subgroups of a group on the structure of the group. Recently, Ballester-Bolinches and Pedraza-Aguilera [5] generalized -quasinormal subgroups to -quasinormally embedded subgroups. is said to be -quasinormally embedded in provided every Sylow subgroup of is a Sylow subgroup of some -quasinormal subgroup of . By applying this concept, Ballester-Bolinches and Pedraza-Aguilera got new criteria for supersolvability of groups. A subgroup of a group is called to be complemented in if has a subgroup such that and . is called to be supplemented in if there exists a subgroup of such that . Obviously, a complemented subgroup is a special supplemented subgroup. In recent years, it has been of interest to use supplementation properties of subgroups to characterize properties of a group. For example, the definition of a weakly -supplemented subgroup introduced by Skiba in [6]. is said to be weakly -supplemented in if has a subgroup such that and , where is the largest -quasinormal subgroup of contained in . Using this concept, many meaningful results have been obtained, such as [7–9]. More recently, the concept of -supplemented subgroups was introduced as follows. Definition 1 (see [10, Definition 1.2]). Let be a subgroup of . We say that is