On intersections of solvable Hall subgroups in finite simple exceptional groups of Lie type

It is proved that, if a Hall subgroup of an almost simple group whose socle is an exceptional group of Lie type is solvable, then there exist four groups conjugate to it whose intersection is trivial.     

On the intersection of conjugate nilpotent subgroups in permutation groups

We prove that, with exception of finitely many explicitly indicated cases, for a nilpotent subgroup in a symmetric or an alternating group there always exists a subgroup conjugate to it such that the intersection of the subgroups is trivial.     

<Emphasis Type="Italic">τ</Emphasis>-primitive subgroups of finite groups

We study the influence of τ-primitive subgroups on the structure of finite groups. As applications, we give the structure of the finite groups whose every subgroup is an intersection of subgroups of prime power index and the structure of finite groups whose every subnormal subgroup is an intersection of subnormal subgroups of prime power index.     

The Influence of Primitive Subgroups on the Structure of Finite Groups

A subgroup $H$ of a group $G$ is said to be primitive if it is a proper subgroup of the intersection of all subgroups of $G$ containing $H$ as its proper subgroup. The purpose of this note is to go further into the influence of primitive subgroups on the structure of finite groups. Some new results are obtained.     

Infinite groups satisfying the normalizer condition for nonprimary subgroups

In this paper we study infinite groups satisfying the normalizer condition for nonprimary subgroups. We show, in particular, that a nonprimary periodic group satisfying this condition is locally finite if the intersection of all its nonprimary subgroups is finite. We establish the local nilpotency of a nonperiodic group satisfying the normalizer condition for nonprimary subgroups. This implies the theorem of S. N. Chernikov which states that a nonperiodic group in which each infinite proper subgroup is different from its normalizer satisfies the normalizer condition.Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 727–734, May, 1976.The author would like to thank D. I. Zaitsev for his assistance and advice during the preparation of this paper.     

Subgroup separability of the figure 8 knot group

It is shown that the fundamental groups of certain non-positively curved 2-complexes have the property that their quasiconvex subgroups are the intersections of finite index subgroups.As a consequence, every geometrically finite subgroup of the figure 8 knot group is the intersection of finite index subgroups. The same result holds for many other prime alternating link groups.     

FC groups whose periodic parts can be embedded in direct products of finite groups

In this note are considered FC groups whose periodic parts can be embedded in direct products of finite groups. It is shown that if the periodic part of an FC group G can be embedded in the direct product of its finite factor groups with respect to the normal subgroups of G whose intersection is the trivial subgroup, then G/Z (G) is a subgroup of a direct product of finite groups. It is also shown that if the periodic part of an FC group G is a group without a center, then G can be embedded in a direct product of finite groups without centers and a torsion-free Abelian group. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 9–20, January, 1977. The author is thankful to the referee for making many valuable remarks.     

On pronormality and strong pronormality of Hall subgroups

We study several well-known questions on pronormality and strong pronormality of Hall subgroups. In particular, we exhibit the examples of finite groups (a) having a Hall subgroup not pronormal in its normal closure (this solves Problem 18.32 of The Kourovka Notebook in the negative); (b) having a Hall subgroup pronormal but not strongly pronormal; and (c) that are simple, having a Hall subgroup, and not strongly pronormal (this solves Problem 17.45(b) of The Kourovka Notebook in the negative).     

The intersection of the subgroups of finite index in Baumslag—Solitar groups

For any one-relator group in the family of Baumslag—Solitar groups, a system of its elements is indicated whose normal closure in the group coincides with the intersection of all normal finite-index subgroups. The well-known criterion for the residual finiteness of Baumslag—Solitar groups is an immediate consequence of this result. It is also shown that, if the intersection of all finite-index normal subgroups in a Baumslag—Solitar group differs from the identity subgroup (i.e., if the group is not residually finite), then this intersection cannot be the normal closure of any finite set of elements.     

Conjugately Dense Subgroups of Locally Finite Chevalley Groups of Lie Rank 1

Of interest are the subgroups of various groups which have nonempty intersection with each class of conjugate elements of the group under study. We call these subgroups conjugately dense and study Neumann's problem of describing them in the Chevalley groups over a field. The main theorem lists all conjugately dense subgroups of the Chevalley groups of Lie rank 1 over a locally finite field.     

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.     

Hall’s Theorem for Limit Groups

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela. Received: May 2006 Revision: May 2007 Accepted: May 2007     

Hall共轭嵌入子群对有限群结构的影响

设群G为有限群,日为G的子群.若对任意的g∈G,日为〈H,H~g〉的Hall子群,则称子群日为G的Hall共轭嵌入子群.利用Hall共轭嵌入子群得到有限群G分别为幂零群与超可解群的若干新的判定方法.     

Finite index subgroups of graph products

We prove that every quasiconvex subgroup of a right-angled Coxeter group is an intersection of finite index subgroups. From this we deduce similar separability results for other types of groups, including graph products of finite groups and right-angled Artin groups.      

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.     

Conjugately dense subgroups of free products of groups with amalgamation

A subgroup having non-empty intersection with each class of conjugate elements of the group is said to be conjugately dense. It is shown that, under certain conditions, the number of conjugately dense subgroups in a free product with amalgamation is not less than some cardinal. As a consequence, P. Neumann’s conjecture in the Kourovka notebook (Question 6.38) is refuted. It is also stated that a modular group and a non-Abelian group of countable or finite rank possess continuum many pairwise non-conjugate conjugately dense subgroups. Supported by RFBR grant No. 03-01-00905. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 520–537, September–October, 2006.     

Finite groups with hall subnormally embedded Schmidt subgroups

Abstract

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.     

Finite groups with Hall subnormally embedded Schmidt subgroups

Abstract

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.     

Tight subgroups in torsion-free Abelian groups

LetX be a torsion-free abelian group. We study the class of all completely decomposable subgroups ofX which are maximal with respect to inclusion. These groups are called tight subgroups ofX and we state sufficient conditions on a subgroup to be tight. In particular we consider tight subgroups of bounded completely decomposable groups. For those we show that every regulating subgroup is tight and we characterize the tight subgroups of finite index in almost completely decomposable groups. The second author was supported by a MINERVA fellowship.     

一类无限群中局部幂零性的判定准则

设G是任意群,群G的Frattini子群nat(G)定义为G的所有极大子群的交.类似地,群G的另外两个特征子群nFrat(G)及R(G)分别定义为群G的所有极大正规子群及群G的所有正规的极大子群的交.本文通过对nat(G),nnat(G)及R(G)的相互包含关系的研究,得到CF-群或中心由多重循环群的扩张群中局部幂零性的一个判定准则.同时也讨论了在某些群类中若干种广义幂零性的等价性.