Sufficient conditions for supersolubility of finite groups

Finite groups with normally embedded subgroups

“…If Q is a quasinormal subgroup of the group G, then Q G =Q G is contained in the hypercenter Z y ðG=Q G Þ of G=Q G . Lemma 2.3 (Ballester-Bolinches and Pedraza-Aguilera [5], [8]). Suppose that U is S-quasinormally (resp.…”

“…Given a group G, two subgroups H and K of G are said to permute if HK ¼ KH, that is, if HK is a subgroup of G. A subgroup H of G is said to be quasinormal in G if it permutes with every subgroup of G. A subgroup H of G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. This concept was introduced by Kegel in 1962 and it has been investigated by many authors; see, for example, [1], [3]- [9], [13], [18]- [23], [25]. Recently, in [5], [8], Ballester-Bolinches and Pedraza-Aguilera extended these concepts to quasinormally and S-quasinormally embedded subgroups. A subgroup H of G is quasinormally (resp.…”

“…Let MðGÞ denote the family of all maximal subgroups of all Sylow subgroups of G. Srinivasan [23] proved that a finite group G is supersolvable if every member of MðGÞ is S-quasinormal in G. This led to the study by many authors of the influence of the members of MðGÞ on the structure of G; see [1], [3]- [9], [15], [21]- [23], [25]. More recently, in [8], Ballester-Bolinches and Pedraza-Aguilera showed that if every member of MðGÞ is S-quasinormally embedded in G, then G is supersolvable. Asaad and Heliel [4] extended this result to saturated formations F containing the class U of all supersolvable groups.…”

On s-quasinormal and c-normal subgroups of a finite group

“…More recently, Ballester-Bolinches and Pedraza-Aguilera [2] generalized S-quasinormal subgroups to S-quasinormally embedded subgroups. A subgroup H of G is said to be S-quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some S-quasinormal subgroup of G. In [2], Ballester-Bolinches and Pedraza-Aguilera showed that if every subgroup in M (G) is S-quasinormally embedded in G, then G is supersolvable. M. Asaad and A.…”

On Generalized Hypercenter of a Finite Group

“…We say, following Kegel [5], that a subgroup of a group G is S -quasinormal in G if it permutes with every Sylow subgroup of G . In 1998, Ballester-Bolinches and Pedraza-Aguilera [3] introduced the following definition: a subgroup H of a group G is said to be S -quasinormally embedded in G if each Sylow subgroup of H is a Sylow subgroup of some S -quasinormal subgroup of G . Obviously, every S -quasinormal subgroup is S -quasinormally embedded.…”