On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

Torsion classes of Specker lattice ordered groups

In this paper we investigate the relations between torsion classes of Specker lattice ordered groups and torsion classes of generalized Boolean algebras.     

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≤Hs¯G, where Hs¯G 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.     

Characterization of modular frobenius groups of special type

In this article, we first investigate the properties of modular Frobenius groups. Then, we consider the case that G' is a minimal normal subgroup of a modular Frobenius group G. We give the complete classification of G when G' as a modular Frobenius kernel has no more than four conjugacy classes in G.     

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.     

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.     

Non-singly-generated multiply ω-fan fitting classes of finite groups

In the paper, we study algebraic lattices of Fitting classes and describe non-singly-generated Fitting classes all of whose proper Fitting subclasses are singly generated. All groups under consideration are assumed finite.     

Finite groups with 2-nilpotent subgroups of even index

Finite groups with 2-nilpotent subgroups of even index are described. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 700–705, May, 1997. Translated by A. I. Shtern     

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 groups with some subgroups of prime indices

We establish the solvability of each finite group whose every proper nonmaximal subgroup lies in some subgroup of prime index.     

A criterion for subnormality of subgroups in finite groups

Based on Wielandt’s criterion for subnormality of subgroups in finite groups, we study 2-maximal subgroups of finite groups and present another subnormality criterion in finite solvable groups.     

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.     

A condition for the solvability of finite groups

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for every maximal subgroup H ₁ of H. We investigate the influence of ?-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite groups.     

Carter subgroups of finite almost simple groups

In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups of every finite group are conjugate. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) for Support of Young Russian Scientists via projects MK-1455.2005.1 and MK-3036.2007.1; by SB RAS Young Researchers Support grant No. 29; via Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007.     

有限群的弱正规子群与p-幂零性

研究了p2阶子群以及一般的pk阶子群为弱正规子群时有限群G的结构.给出了有限群为p-幂零群以及超可解群的一些条件.     

某些C-正规子群对有限群结构的影响

利用某些子群的C正规性,得到有限群成为可解的一系列充分条件