The Subgroups of Direct Products of Surface Groups

A subgroup of a product of n surface groups is of type FP _n if and only if it contains a subgroup of finite index that is itself a product of (at most n) surface groups.     

Bounded Rank Subgroups of Coxeter Groups, Artin Groups and One-Relator Groups with Torsion

We obtain a number of results regarding the freeness of subgroupsof Coxeter groups, Artin groups and one-relator groups withtorsion. In the case of Coxeter groups, we also obtain resultson quasiconvexity and subgroup separability. 2000 MathematicsSubject Classification 20F65, 20F55, 20F36, 20F06.     

Automorphisms of Free Groups and Outer Space

This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its variations.     

On the width in free products with amalgamation

In this paper, the problem of finite width for free products of groups with amalgamated subgroup is considered. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 353–359, September, 2000.     

关于Weyl群的某些研究

记 Φ为欧氏空间 V中某不可约根系 ,具有 Weyl群 W,记 σ为 W中满足条件 w( Φ+ ) =Φ-的唯一元 .本文考虑如何将 σ分解成反射之积 ;σ在 Φ上的作用方式如何 .作为应用确定了 W的中心 ;进一步确定了 V的一类子空间在 W中的固定子群 .     

关于“L—Fuzzy群同态与同构及其性质”一文的注记

文（１）给出了Ｌ－Ｆｕｚｚｙ群间的同态映射，满同态映射以及Ｌ－Ｆｕｚｚｙ群的Ｌ－正规子群等概念。本文指出以上三种定义的不合理性，并给出它们的合理性定义。     

On the definability of endomorphisms of a free group of the variety\mathfrak{A}\mathfrak{M} by a finite set of valuesby a finite set of values

In the paper we find out in what cases any endomorphism of a free metabelian group of rankn is uniquely determined by its values on finitely many elements of the group. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 916–920, December, 1997 Translated by A. I. Shtern     

On G-Covering Subgroup Systems for Some Saturated Formations of Finite Groups

Let ? be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for ? if G ∈ ? whenever Σ ? ?. For a non-identity subgroup H of G, we put Σ _H be some set of subgroups of G which contains at least one supplement in G of each maximal subgroup of H. Let p ≠ q be primes dividing |G|, P, and Q be non-identity a p-subgroup and a q-subgroup of G, respectively. We prove that Σ _P and Σ _P  ∪ Σ _Q are G-covering subgroup systems for many classes of finite groups.     

Howson's Property for Semidirect Products of Semilattices by Groups

An inverse semigroup S is a Howson inverse semigroup if the intersection of finitely generated inverse subsemigroups of S is finitely generated. Given a locally finite action θ of a group G on a semilattice E, it is proved that E*_θG is a Howson inverse semigroup if and only if G is a Howson group. It is also shown that this equivalence fails for arbitrary actions.     

On Asymptotic Dimension of Groups Acting on Trees

We prove the following.THEOREM. Let be the fundamental group of a finite graph of groups with finitely generated vertex groups G _v having asdim G _v n for all vertices v. Then asdim n+1.This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.     

A Characterization of Root Classes of Groups

We prove that a class of groups is root in a sense of K. W. Gruenberg if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we obtain a condition which is sufficient for the generalized free product of two nilpotent groups to be residual solvable.     

On the Validity of Thompson's Conjecture for Finite Simple Groups

In this article, we prove a conjecture of J. G. Thompson for the finite simple group ² D _n (q). More precisely, we show that every finite group G with the property Z(G) = 1 and N(G) = N(² D _n (q)) is necessarily isomorphic to ² D _n (q). Note that N(G) is the set of lengths of conjugacy classes of G.     

On Some Functional Equations and Representations of Topological Semigroups

Let a representation T of a semigroup G on a linear space X be given. We call x ∈ X a finite vector if its orbit T(G) is contained in a finite-dimensional subspace. In this paper, some statements about finite vectors are applied to the following problem. For a given positive integer n > 1, describe all continuous functions f : G → ? such that the function (x₁,..., x _n ) ? f(x₁ + ? + x _n ) can be polynomially expressed via functions of sums of fewer variables.     

On the Solvability of Finite Irreducible Linear Groups with Hall TI-Subgroups

Conditions for a -solvable complex linear group of a relatively small degree to be solvable are found.     

关于素数阶循环群中短序列的等价类

考虑了素数阶循环群中的短序列的等价序列,并在某些情况下给出序列的Index值的上界．     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.     

有限$p$-幂零群的一个判定准则

设$G$为一个有限群, $H$是$G$的一个子群. 称$H$在$G$中是$s$-半置换的若对$G$的任意Sylow $p$-子群$G_p$, $HG_p=G_pH$, 其中$(p, |H|)= 1$,这里$p$是整除$G$的阶一个素数.通过假设$G$的一些子群是$s$-半置换的, 我们给出了$p$-幂零群的一个判定准则. 我们的结果推广了著名的Burnside $p$-幂零群准则.