Groups with many subgroups having polycyclic-by-finite layers or derived subgroup

A group is called (PF)L if the subgroups generated by its elements having same order (finite or infinite) are polycyclic-by-finite. In the present paper we prove that a group is locally graded minimal non-((PF)L∪(𝔓𝔉)𝔄) if, and only if, it is non-perfect minimal non-FC, where (𝔓𝔉)𝔄 denotes the class of (polycyclic-by-finite)-by-abelian groups. We prove also that a group of infinite rank whose proper subgroups of infinite rank are in ((PF)L∪(𝔓𝔉)𝔄) is itself in ((PF)L∪(𝔓𝔉)𝔄) provided that it is locally (soluble-by-finite) without simple homomorphic images of infinite rank. Our last result concerns groups that satisfy the minimal condition on non-((PF)L∪(𝔓𝔉)𝔄)-subgroups.     

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 Weakly Factorizable Groups

Groups with complemented subgroups, which are also called completely factorizable groups, were studied by P. Hall, S. N. Chernikov, and N. V. Chernikova (Baeva). For complete factorizability, it is sufficient (Theorem 1) that each proper subgroup have a normal complement in some larger subgroup. A group is said to be weakly factorizable if each of its proper subgroups is complemented in some larger subgroup; the problem of describing finite groups with this property is posed (Question 8.31) in the Kourovka Notebook. Some properties of these groups are considered. The question is studied for Sylow p-subgroups of Chevalley-type groups of characteristic p. The main theorem, Theorem 2, establishes the weak factorizability of the Sylow p-subgroups in the symmetric and alternative groups and in the classical linear groups over fields of characteristic p> 0, excluding the unitary groups of odd dimension > p.     

Finite groups acting 2-transitively on their sylow subgroups

In this paper we have completely determined: (1) all almost simple groups which act 2-transitively on one of their sets of Sylow p-subgroups. (2) all non-abelian simple groups T whose automorphism group acts 2-transitively on one of the sets of Sylow p-subgroups of T. (3) all finite groups which are 2-transitive on all their sets of Sylow subgroups. The first author acknowledges the support of OPR Scholarship of Australia The second author is supported by the National Natural Science Foundation of China. Thanks are also due to the Department of Mathematics, the University of Western Australia, where he did his part of this work for its hospitality     

Locally minimal topological groups 2

We continue in this paper the study of locally minimal groups started in Außenhofer et al. (2010) [4]. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group G almost minimal if it has a closed, minimal normal subgroup N such that the quotient group G/N is uniformly free from small subgroups. The class of almost minimal groups includes all locally compact groups, and is contained in the class of locally minimal groups. On the other hand, we provide examples of countable precompact metrizable locally minimal groups which are not almost minimal. Some other significant properties of this new class are obtained.     

The Existence of p-Blocks with a Submaximal-Subgroup of a Sylow p-Subgroup as Defect Group

Abstract

In this paper,we will determine the existence of blocks with submaximal subgroups of Sylow p-subgroups of a finite group G as defect groups.     

The Influence of Minimal Subgroups on the Structure of Finite Groups

Abstract Let G be a finite group. The question how the properties of its minimal subgroups influence the structure of G is of considerable interest for some scholars. In this paper we try to use c-normal condition on minimal subgroups to characterize the structure of G. Some previously known results are generalized. The author is supported in part by NSF of China and NSF of Guangdong Province     

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.     

Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

Nearly m-embedded subgroups and solvability of finite groups

A subgroup A of a finite group G is called {1≤G}-embedded in G if for each two subgroups K≤H of G, where K is a maximal subgroup of H, A either covers the pair (K,H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1≤G}-embedded subgroup C such that G?=?HT and H∩T≤C≤H. In this paper, we mainly prove that G is solvable if and only if its Sylow 3-subgroups, Sylow 5-subgroups and Sylow 7-subgroups are nearly m-embedded in G.     

Bounds in the Theory of Finite Covers

We give an upper bound for the number of conjugacy classes of closed subgroups of the full wreath product FWr_WSym(Ω) which project onto Sym(Ω). Here, Ω is infinite, W is the set of n-tuples of distinct elements from Ω (for some finite n), F is a finite nilpotent group, and the topology on the wreath product is that of pointwise convergence in its imprimitive permutation action. The result addresses a problem which arises in a natural model-theoretic context about classifying certain types of finite covers.     

两个正规p-超可解子群的积

本文研究了两个正规的p-超可解子群的积构成的极小非P-超可解群的结构的问题.利用有限群论和群类论的一些基本方法,获得了两个正规的p-超可解子群的积仍为P-超可解群的一些充分条件,并推广了文[1]中关于超可解群的情况.     

On minimal non-MSN-groups

A finite group G is called an MSN-group if all maximal subgroups of the Sylow subgroups of G are subnormal in G. In this paper, we determinate the structure of non-MSN-groups in which all of whose proper subgroups are MSN-groups.     

A Generalization of Cover-Avoiding Properties in Finite Groups

A subgroup H of a group G is called a CAP*-subgroup of G, if H either covers or avoids every non-Frattini chief factor of G. In this article, we give some interesting properties of CAP*-subgroups. Furthermore, we determine the structure of finite groups based on the assumption that some subgroups are CAP*-subgroups and generalize some known results.     

On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups

Baer and Wielandt in 1934 and 1958, respectively, considered that the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this article, for a finite group G, we define the subgroup S(G) to be intersection of the normalizers of all non-cyclic subgroups of G. Groups whose noncyclic subgroups are normal are studied in this article, as well as groups in which all noncyclic subgroups are normalized by all minimal subgroups. In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups.     

Relative minimality and co-minimality of subgroups in topological groups

We study two properties of subgroups of a topological group (relative minimality and co-minimality), that generalize minimality. Many applications, mostly related to semidirect products and generalized Heisenberg groups are given.     

Finite edge-transitive Cayley graphs and rotary Cayley maps

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups.

In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

     

On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if N _G (H) ∩ H ^x  ≤ H for all x ∈ G. A subgroup H of G is called weakly ?-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ?-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ?-subgroups in G. Some recent results are extended and generalized.     

On minimal non- -groups

A finite group G is called an J N J-group if every proper subgroup H of G is either subnormal in G or self-normalizing. We determinate the structure of non-J N J-groups in which all proper subgroups are J N J- groups.     

Influence of certain permutable subgroups on finite smooth groups

A subgroup H of a finite group G is said to be permutable in G if it permutes with every subgroup of G. In this paper, we determine the finite groups which have a permutable subgroup of prime order and whose maximal subgroups are totally (generalized) smooth groups.