On generalized m-S-permutable subgroups of a finite group

In this paper, we give some new conditions under which a normal subgroup E of a finite group G is hypercyclically embedded in G, that is, every chief factor of G below E is cyclic.     

Finite groups with S-permutable n-minimal subgroups

We study the supersolvability of finite groups and the nilpotent length of finite solvable groups under the assumption that all their exactly n-minimal subgroups are S-permutable, where n is an arbitrary integer.     

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 the n-minimal subgroups

Let G be a finite group and n be a positive integer. An n-minimal subgroup H of G is called to be exactly n-minimal if no proper subgroup of H is n-minimal. In this paper, we study the solvability of G under the assumption that all exactly n-minimal subgroups of G are S-permutable.     

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.     

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.     

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.     

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 H_se of G contained in H such that G = HT and H∩T≤H_se. 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 p-nilpotency of finite groups with some subgroups π-quasinormally embedded

Summary A subgroup H of a group G is said to be π-quasinormal in G if it permutes with every Sylow subgroup of G, and H is said to be π-quasinormally embedded in G if for each prime dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some π-quasinormal subgroups of G. We characterize p-nilpotentcy of finite groups with the assumption that some maximal subgroups, 2-maximal subgroups, minimal subgroups and 2-minimal subgroups are π-quasinormally embedded, respectively.     

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.     

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.     

A characterization of the finite simple groups with set of indices of their maximal subgroups

We study the problem concerning the influence of indices of maximal subgroups of a simple group on the structure of a group. We obtain a characterization property of all finite simple groups.     

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.     

Carter subgroups of finite groups

It is proven that the Carter subgroups of a finite group are conjugate. A complete classification of the Carter subgroups in finite almost simple groups is also obtained.     

次正规子群与有限群的超可解性

通过Sylow子群的极大子群和次正规性,利用极小阶反例的方法,得出群p-幂零性和超可解性的结论.本文的创新改进之处在于结合Sylow子群的极大子群和次正规性,研究p-幂零性和超可解性的相关结论.     

Semipermutable subgroups and s-semipermutable subgroups in finite groups

Suppose that H is a subgroup of a finite group G. We call H is semipermutable in G if HK = KH for any subgroup K of G such that (|H|, |K|) = 1; H is s-semipermutable in G if HG_p = G_pH, for any Sylow p-subgroup G_p of G such that (|H|, p) = 1. These two concepts have been received the attention of many scholars in group theory since they were introduced by Professor Zhongmu Chen in 1987. In recent decades, there are a lot of papers published via the application of these concepts. Here we summarize the results in this area and gives some thoughts in the research process.     

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