On Generalized PST-groups

A finite group G is called a generalized PST-group if every subgroup contained in F（G） permutes all Sylow subgroups of G, where F（G） is the Fitting subgroup of G. The class of generalized PST-groups is not subgroup and quotient group closed, and it properly contains the class of PST-groups. In this paper, the structure of generalized PST-groups is first investigated. Then, with its help, groups whose every subgroup （or every quotient group） is a generalized PST-group are deter- mined, and it is shown that such groups are precisely PST-groups. As applications, T-groups and PT-groups are characterized.

Finite Groups with Certain S-Permutable and GS-Maximal Subgroups

Let G be a finite group and H a subgroup of G.We say that H is S-permutable in G if H permutes with every Sylow subgroup of G.A group G is called a generalized smooth group(GS-group)if[G/L]is totally smooth for every subgroup L of G of prime order.In this paper,we investigate the structure of G under the assumption that each subgroup of prime order is S-permutable if the maximal subgroups of G are GS-groups.

On weakly s-permutably embedded subgroups of finite groups （II）

Suppose that G is a finite group and H is a subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing ｜H｜, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup Hse of G contained in H such that G = HT and H n T ≤ Hse. In this paper, we continue the work of [Comm. Algebra, 2009, 37： 1086-1097] to study the influence of the weakly s-permutably embedded subgroups on the structure of finite groups, and we extend some recent results.

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgroup F*(G) of G, at least one of the following holds: (1) Every maximal subgroup of P is S-embedded in G. (2) Every cyclic subgroup H of P with prime order or order 4 (if P is a non-abelian 2-group and H Z∞(G)) is S-embedded in G.

S-semiembedded subgroups of finite groups

A subgroup H of a finite group G is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. We say that a subgroup H of a finite group G is S-semiembedded in G if there exists an s-permutable subgroup T of G such that TH is s-permutable in G and T ∩ H ≤ H-sG, where HsG is an s-semipermutable subgroup of G contained in H. In this paper, we investigate the influence of S-semiembedded subgroups on the structure of finite groups.

Some Criteriafor p-SupersolvabilityofaFiniteGroup

A subgroup H of a finite group G is said to be S-semipermutable in G if H permutes with all Sylow q-subgroups of G for the primes q not dividing the order of H.Some criteria for p-supersolvability of a finite group are given,which are the generalizations of many recent results.