A Note on s-quasinormally Embedded and c-supplemented Subgroups of Finite Groups

In this paper the influence of s-quasinormally embedded and c-supplemented subgroups on the p-nilpotency of finite groups is investigate and some recent results are generalized.

On s-Quasinormal and c-Normal Subgroups of a Finite Group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.

On SS-quasinormal subgroups and the structure of finite groups

A subgroup H of a finite group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this paper, we investigate the structure of a group under the assumption that every subgroup with order pm of a Sylow p-subgroup P of G is SS-quasinormal in G for a fixed positive integer m. Some interesting results related to the p-nilpotency and supersolvability of a finite group are obtained. For example, we prove that G is p-nilpotent if there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| or 2|D| whenever p = 2 and |D| = 2 is SS-quasinormal in G, where p is the smallest prime dividing the order of G and P is a Sylow p-subgroup of G.

On p-Cover-Avoid and S-Quasinormally Embedded Subgroups in Finite Groups

Let G be a finite group, p the smallest prime dividing the order of G and P a Sylow p-subgroup of G. If d is the smallest generator number of P, then there exist maximal subgroups P1, P2,..., Pd of P, denoted by Md（P） = {P1,...,Pd}, such that di=1 Pi = Φ（P）, the Frattini subgroup of P. In this paper, we will show that if each member of some fixed Md（P） is either p-cover-avoid or S-quasinormally embedded in G, then G is p-nilpotent. As applications, some further results are obtained.

On Nearly SS-Embedded Subgroups of Finite Groups

Abstract Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H∩ K≤HseG, where HseG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.

Onc*-Normal Subgroups in Finite Groups

A subgroup H of a finitegroup G is called a c＊-normal subgroup of G if there exists a