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.     

Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.     

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.     

A characterization of finite simple groups by the orders of solvable subgroups

Let G be a finite group and S be a finite simple group. In this paper, we prove that if G and S have the same sets of all orders of solvable subgroups, then G is isomorphic to S, or G and S are isomorphic to Bn(q), Cn(q), where n≥3 and q is odd. This gives a positive answer to the problem put forward by Abe and Iiyori.     

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.     

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     

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.     

Finite groups with abelian second maximal subgroups

Abstract

In this article, we give a complete classification of finite groups whose second maximal subgroups are all abelian.     

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 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.     

Congruence subgroups of Hecke groups

Hecke groups are an important tool in subgroups of Hecke groups play an important rule investigating functional equations, and congruence in research of the solutions of the Dirichlet series. When q, m are two primes, congruence subgroups and the principal congruence subgroups of level m of the Hecke group H（√q） have been investigated in many papers. In this paper, we generalize these results to the case where q is a positive integer with q ≥ 5, √q ￠ Z and m is a power of an odd prime.     

n-极大子群为共轭可换的有限群

群G的子群H称为G的共轭可换子群,若HHg=HgH,对任意g∈G都成立,本文考查了n-极大子群为共轭可换时对有限群构造的影响.     

Solitary subgroups of Abelian groups

Since solitary subgroups of (infinite) Abelian groups are precisely the strictly invariant subgroups which are co-Hopfian (as groups), and strictly invariant subgroups turn out to be strongly invariant for large classes of Abelian groups we determine the solitary subgroups for these classes of groups.