
Algebra 2013
Finite Groups with Some Supplemented SubgroupsDOI: 10.1155/2013/153691 Abstract: 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, BallesterBolinches and PedrazaAguilera [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, BallesterBolinches and PedrazaAguilera 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
