On P𝔉-Supplemented subgroups of finite groups

Let 𝔉 be a class of groups and G a finite group. A maximal subgroup M of G is called 𝔉-abnormal provided G∕M_G?𝔉. Let K<H be subgroup of G. Then we say that (K,H) is an 𝔉-abnormal pair of G provided K is a maximal 𝔉-abnormal subgroup of H. Let A be a subgroup of G. Then we say that A is 𝔉-quasipermutable in G provided A either covers or avoids every 𝔉-abnormal pair of G. In this paper, we consider some applications of 𝔉-quasipermutable subgroups.     

On the 𝔉‐hypercentre of a finite group

The main objective of this paper is to study and describe the hypercentre of a finite group associated with saturated formations, in terms of some subgroup embedding properties related to permutability. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On g-s-supplemented subgroups of finite groups

A subgroup H of a group G is said to be g-s-supplemented in G if there exists a subgroup K of G such that HK⊴G and H ∩ K ⩽ H _sG , where H_sG is the largest s-permutable subgroup of G contained in H. By using this new concept, we establish some new criteria for a group G to be soluble.     

On two classes of finite supersoluble groups

Let ? be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ?-S-semipermutable if H permutes with every Sylow p-subgroup of G in ? for all p?π(H); H is said to be ?-S-seminormal if it is normalized by every Sylow p-subgroup of G in ? for all p?π(H). The main aim of this paper is to characterize the ?-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ? are ?-S-semipermutable in G and the ?-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ? are ?-S-seminormal in G.     

On finite soluble groups in which Sylow permutability is a transitive relation

A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved. The key point is an extension of a subnormality criterion due to Wielandt. This revised version was published online in June 2006 with corrections to the Cover Date.     

On s-semipermutable subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. We say that H is ssemipermutable in G if HGv = GpH for any Sylow p-subgroup Gp of G with （p, ｜H｜） = 1. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

Lie Algebra 𝔉-Normalizers are Intravariant

Let 𝔉 be a saturated formation of soluble Lie algebras. Let L be a soluble Lie algebra, and let U be an 𝔉-normalizer of L. Then U is intravariant in L.     

S-quasinormality of finite groups

Let d be the smallest generator number of a finite p-group P, and let be a set of maximal subgroups of P such that . In this paper, the structure of a finite group G under some assumptions on the S-quasinormally embedded or SS-quasinormal subgroups in , for each prime p, and Sylow p-subgroups P of G is studied.     

A solvability criterion for finite groups

We show that if for every prime p, the normalizer of a Sylow p-subgroup of a finite group G admits a p-solvable supplement, then G is solvable. This generalizes a solvability criterion of Hall which asserts that a finite group G is solvable if and only if G has a Hall p′-subgroup for every prime p.     

On the structure of generalized symmetric spaces of SLn𝔽q)

In this paper we extend previous results regarding SL₂(k) over any finite field k by investigating the structure of the symmetric spaces for the family of special linear groups SL_n(k) for any integer n>2. Specifically, we discuss the generalized and extended symmetric spaces of SL_n(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal.