Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

群的融合自由积的πn-Frattini子群和πc-Frattini子群

Azarian M K将Tang C Y得到的关于两个群的带循环融合自由积的Frattini子群的一个定理推广到任意多个子群的带循环融合自由积的情况.通过考虑任意多个子群的带循环融合自由积的πnFrattini子群和πcFrattini子群,得到了类似的结果.     

On the normal indices of proper subgroups of finite groups

We extended the normal index from maximal subgroups to proper subgroups. We give a quantitative version of all results obtained by using c-normal subgroups and obtain some new characterizations of solvable, supersolvable and nilpotent groups by the normal indices of proper subgroups.     

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.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

On <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript>-supplemented subgroups of finite groups

Suppose that F is a formation of finite groups. We introduce the concept of F _h -supplemented subgroups and investigate the structure of finite groups on assuming that some maximal subgroups of Sylow subgroups, maximal subgroups, minimal subgroups, and 2-maximal subgroup are F _h -supplemented, respectively. Some available results are generalized.     

Frattini subgroups and supernilpotent groups

In the context of the problem of which nonabelianp-groups can occur as normal subgroups contained in Frattini subgroups, the family of supernilpotent groups (all maximal subgroups characteristic) is investigated. Results of this investigation are applied to the Frattini-embedding problem, incorporating recent work of A. R. Makan. The groups of order 2ⁿ (n ≦ 6) have been examined with respect to supernilpotence and their occurrence as normal subgroups contained in Frattini subgroups. Results of this examination are presented.     

Large Abelian Unipotent Subgroups of Finite Chevalley Groups

We compute maximal orders of unipotent Abelian subgroups, estimate p-ranks, and describe the structure of Thompson subgroups of maximal unipotent subgroups of finite exceptional groups of Lie type.     

Parabolic subgroups in FC-type Artin groups

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results of Cumplido, Gebhardt, Gonzales-Meneses and Wiest, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC-type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed by Cumplido et al. using parabolic subgroups. We extend the construction of this complex, called the complex of parabolic subgroups, to FC-type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ-hyperbolic.     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.     

On subgroups of finite index in compact Hausdorff groups

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.     

On Influence of Contranormal Subgroups on the Structure of Infinite Groups

Following Rose, a subgroup H of a group G is called contranormal, if G = H ^G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.     

X-permutable maximal subgroups of Sylow subgroups of finite groups

We study finite groups whose maximal subgroups of Sylow subgroups are permutable with maximal subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1299–1309, October, 2006.     

The Depth of Subgroups of Suzuki Groups

We determine the combinatorial depth of certain subgroups of simple Suzuki groups Sz(q), among others, the depth of their maximal subgroups. We apply these results to determine the ordinary depth of these subgroups.     

Finite two-are transitive graphs admitting a ree simple group

ABSTRACT

The article is dedicated to some generalizations of minimax soluble groups satisfying common criterion of nilpotency, such that normality of maximal subgroups, nilpotency of the factor-group by the Frattini subgroup, normality of pronormal subgroups, non-existence of proper abnormal subgroups and so on.     

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.     

Embedding of Nonprimary Subgroups in the Symmetric Group S 9

The nonprimary subgroups of the symmetric group S ₉ are investigated. Embedding properties of these subgroups are listed in a table. Properties such as abnormality, pronormality, paranormality, weak normality, etc. were checked with the help of a computer. Algorithms and codes of the first author were used for this purpose. The research leans upon the technique of Burnside marks, as well as upon pertinent information on the table of marks of S ₉ from the computer algebra package GAP. The subgroups were investigated up to conjugacy; the total number of conjugacy classes of nonprimary subgroups of S ₉ is 432. Some subgroups were additionally checked by other programs based on the double coset method. Bibliography: 24 titles.