Commutative Group Algebras of Summable p-Groups

We study numerical semigroups generated by generalized arithmetic sequences. We present a membership criterion for such a numerical semigroup, and by this we are able to answer fundamental questions concerning a numerical semigroup such as computing the Frobenius number and the type of the numerical semigroup, and decide whether the numercial semigroup is symmetric. Also for this kind of numerical semigroups, we compute the cardinality of a minimal presentation and determine whether they are complete intersections.     

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.      

Notes on Essentially Finitely Indecomposable Nonthick Primary Abelian Groups

Let R be a commutative Noetherian ring of prime characteristic and M be an x-divisible right R[x, f]-module that is Noetherian as R-module. We give an affirmative answer to the question of Sharp and Yoshino in the case where R is semilocal and prove that the set of graded annihilators of R[x, f]-homomorphic images of M is finite. We also give a counterexample in the general case.     

Double Coset Decompositions of Groups

We show that residually finite or word hyperbolic groups which can be decomposed as a finite union of double cosets of a cyclic subgroup are necessarily virtually cyclic, and we apply this result to the study of Frobenius permutation groups. We show that in general, finite double coset decompositions of a group can be interpreted as an obstruction to splitting a group as a free product with amalgamation or an HNN extension.     

Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product. Received: January 2006, Revision: May 2006, Accepted: May 2006     

Auslander–Reiten Quivers and the Coxeter Complex

Let Q be a quiver of type ADE. We construct the corresponding Auslander–Reiten quiver as a topological complex inside the Coxeter complex associated with the underlying Dynkin diagram. In A_n case, we recover special wiring diagrams. Presented by R. RentschlerMathematics Subject Classifications (2000) 16G70, 17B10, 20F55.     

Subgroup Separability and Linearity of Certain Graphs of Groups

We show that if the fundamental group of a finite graph of groups with finitely generated Abelian vertex groups is subgroup separable, then it is linear.     

Sur les immeubles hyperboliques (On hyperbolic buildings)

The aim of this paper is to give a geometric approach to Tits' amalgam method to construct buildings and to initiate a study of hyperbolic buildings, i.e. whose types are reflexion systems of the real hyperbolic space. We construct lots of examples and study their cohomology at infinity. We construct CAT(–1) polyhedral complexes having big discrete parabolic groups of isometries.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Let p₁, p₂, p₃ be primes. This is the final paper in a series of three on the (p₁, p₂, p₃)-generation of the finite projective special unitary and linear groups PSU ₃(pⁿ), PSL ₃(pⁿ), where we say a noncyclic group is (p₁, p₂, p₃)-generated if it is a homomorphic image of the triangle group T_{p1, p2, p3} . This article is concerned with the case where p₁ = 2 and p₂ ≠ p₃. We determine for any primes p₂ ≠ p₃ the prime powers pⁿ such that PSU ₃(pⁿ) (respectively, PSL ₃(pⁿ)) is a quotient of T = T_{2, p2, p3} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU ₃(pⁿ)) (respectively, Hom(T, PSL ₃(pⁿ))) is surjective as pⁿ tends to infinity.     

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

Counting Overlattices in Automorphism Groups of Trees

We give an upper bound for the number u _Γ(n) of “overlattices” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a 2p-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well.     

PSL（2，F）的一个嵌入定理及其应用

设Ｆ是任意域，Ｇ代表ＳＬ（２，Ｆ）或ＰＳＬ（２，Ｆ）．本文的主要结果是：设Ｋ是Ｆ的子域，则Ｇ中同构于ＳＬ（２，Ｋ）或ＰＳＬ（２，Ｋ）的子群在Ｇ的自同构的作用下彼此共轭，利用这一结果，本文明确确定了A₁^[1]型的仿射Ｋａｃ－Ｍｏｏｄｙ群的一类极大正规子群．     

Maximal covers of finite groups

Abstract

Let λ(G) be the maximum number of subgroups in an irredundant covering of the finite group G. We prove that if G is a group with λ(G) ≤ 6, then G is supersolvable. We also describe the structure of groups G with λ(G) = 6. Moreover, we show that if G is a group with λ(G)?<?31, then G is solvable.     

On Finite Groups with Few Automorphism Orbits

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A₅, A₆, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A₅.     

The equation x 2 y 2 = g in partially commutative groups

A partially commutative group is a group defined by generators and relations so that all defining relations are of the form: the commutator of two generators is equal to the identity element. We consider an algorithm for checking whether a given group element is a product of two squares. This generalizes a result of Wicks for free groups.     

Decomposition theorems for groups of diffeomorphisms in the sphere

We study the algebraic structure of several groups of differentiable diffeomorphisms in . We show that any given sufficiently smooth diffeomorphism can be written as the composition of a finite number of diffeomorphisms which are symmetric under reflection, essentially one-dimensional and about as differentiable as the given one.

     

Finite Soluble Group Generated by the Conjugacy Class of an Involution

Let G be a finite group and P a subgroup of order 2. We study in this article the structures of the soluble subgroup of G that is generated by three conjugates of P. We use the results we proved about the soluble subgroups that are generated by three conjugates of P to find a soluble analogue of the Baer–Suzuki Theorem in the case prime 2.     

Groups with exactly two supercharacter theories

In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are ?₃ and S₃. We also show that the only nonsolvable group with two supercharacter theories is Sp(6,2).     

Symmetric Presentations for the Fischer Groups II: The Sporadic Groups

We give symmetric presentations for groups closely related to the three sporadic Fischer groups Fi₂₂, Fi₂₃ and Fi₂₄.Mathematics Subject Classiffications (2000). 20D08, 20D06, 20F05This research was supported by an EPSRC grant ref. GR/N27491/01     

Effective Indices of Subgroups in Baumslag–Pride Groups with Free Quotients

Given a group with at least two more generators than relations, we give an effective estimate on the minimal index of a subgroup with a nonabelian free quotient. We show that the index is bounded by a polynomial in the length of the relator words. We also provide a lower bound on the index.