Γ-amenability of locally compact groups

Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to F-amenable groups and give various characterizations for F-amenable groups. A sequence of characterizations of F-amenable groups is given here by developing the well-known Flner＇s conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

On generalized fibonacci groups with an odd number of generators

The geometrical properties of cyclically presented groups of Fibonacci type F(r,m, k)and H(r,m,k)are discussed. It is shown that for even rand odd m some infinite family of generalized Fibonacci groups F(r, m, k)cannot be fundamental groups of hyperbolic 3-orbifolds of finite volume.     

A minimal Cantor set in the space of 3-generated groups

We construct and study a family of 3-generated groups parametrized by infinite binary sequences w. We show that two groups of the family are isomorphic if and only if the sequences are cofinal and that two groups cannot be distinguished by finite sets of relations. We show a connection of the family with 2-dimensional holomorphic dynamics.      

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

Some Properties of Bell Groups

For any integer n ≠ 0,1, a group G is said to be “n-Bell” if it satisfies the identity [x ⁿ ,y] = [x,y ⁿ ]. It is known that if G is an n-Bell group, then the factor group G/Z ₂(G) has finite exponent dividing 12n ⁵(n ? 1)⁵. In this article we show that this bound can be improved. Moreover, we prove that every n-Bell group is n-nilpotent; consequently, using results of Baer on finite n-nilpotent groups, we give the structure of locally finite n-Bell groups. Finally, we are concerned with locally graded n-Bell groups for special values of n.     

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

On the Hopficity of HNN-Extensions of Polycyclic Groups

For a group G, an HNN-extension of polycyclic groups, we give two necessary and sufficient conditions for G to be hopfian. One is based on the nature of the endomorphism of G and the other on the nature of the associated subgroups of G. At the end, we give an application for HNN-extensions of nilpotent groups.     

Non-Hopfian Relatively Free Groups

To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960’s, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-Hopfian.     

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.     

On Groups with the Second Largest Value of the Sum of Element Orders

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group ? _n . In this paper, we investigate groups with the second largest value of ψ on the set of groups of the same order.     

3-Rewritable Nilpotent 2-Groups of Class 2#

ABSTRACT

Let n be an integer greater than 1. A group G is said to be n-rewritable (or a Q_n-group) if for every n elements x₁, x₂,…,x_n in G there exist distinct permutations σ and τ in S_n such that x_σ(1)x_{σ (2)} ??? x_σ(n) = x_τ(1)x_τ(2) ??? x_τ(n). In this paper, we characterize all 3-rewritable nilpotent 2-groups of class 2. Also we have found a bound for the nilpotency class of certain nilpotent 3-rewritable groups, and have shown that 3-rewritable groups satisfy a certain law.     

Characterizing Translations on Groups by Cosets of Their Subgroups

On a group, constant functions and left translations by group elements map left cosets into left cosets for every subgroup. We determine classes of groups for which this property of preserving cosets characterizes constants and translations, e.g., finite non-abelian groups that are perfect, partitioned, primitive, or generated by elements of prime order p. For certain classes of groups we construct other coset-preserving functions, in particular, power endomorphisms and functions defined in terms of the subgroup lattice.     

Elementary Classes of Groups

Let be a class of groups. The elementary class with base is defined as the minimal class of groups containing and closed with respect to taking subgroups, quotient groups, group extensions, and direct limits. Properties of such classes are studied. Some applications to the theory of elementary amenable groups and a relation to the Kurosh--Chernikov classes of generalized solvable groups are considered.     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

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.     

Two-parameter families of quantum symmetry groups

We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of non-negative integers we define and investigate quantum groups O⁺(p,q), B⁺(p,q), S⁺(p,q) and H⁺(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of ) free quantum groups studied earlier. For H⁺(p,q) the situation is different and we show that , where the latter can be viewed as a liberation of the classical isometry group of the p-dimensional torus.     

Bestvina's Normal Form Complex and the Homology of Garside Groups

A Garside group is a group admitting a finite lattice generating set . Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice , and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvina's weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually Abelian.     

Large localizations of finite groups

We construct examples of localizations in the category of groups which take the Mathieu group M₁₁ to groups of arbitrarily large cardinality which are “abelian up to finitely many generators.” The paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations.