Projective Resolutions for Modules over Infinite Groups

We define a notion of complexity for modules over group rings of infinite groups. This generalizes the notion of complexity for modules over group algebras of finite groups. We show that if M is a module over the group ring kG, where k is any ring and G is any group, and M has f-complexity (where f is some complexity function) over some set of finite index subgroups of G, then M has f-complexity over G (up to a direct summand). This generalizes the Alperin-Evens Theorem, which states that if the group G is finite then the complexity of M over G is the maximal complexity of M over an elementary abelian subgroup of G. We also show how we can use this generalization in order to construct projective resolutions for the integral special linear groups, SL(n, ℤ), where n ≥ 2.     

On a Conjecture of Groves for Modules Over Infinite Nilpotent Groups

We show that a conjecture of Groves for modules over nilpotent groups of class 2 holds for the codimension 2 case with certain assumptions.     

Groups whose Proper Subgroups of Infinite Rank Have a Transitive Normality Relation

A group G is called a T-group if all its subnormal subgroups are normal, and G is a ${\bar{T}}$ -group if every subgroup of G has the property T. It is proved here that if G is a locally soluble group whose proper subgroups of infinite rank have the T-property, then either G is a ${\bar{T}}$ -group or it has finite rank.     

Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.     

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).     

Generating Infinite Symmetric Groups

Let S = Sym(     

Abelian Groups and Regular Modules

The structure of the additive group of a regular module is considered. Abelian groups that are regular modules over their rings of endomorphisms are studied. Nonreduced endoregular groups and endoregular torsion-free groups of finite rank are described.     

Periodic Modules and Quantum Groups

We prove that the elements A_\leqslant defined by Lusztig in a completion of the periodic module actually live in the periodic module (in the type A case). In order to prove this, we compare, using the Schur duality, these elements with the Kashiwara canonical basis of an integrable module.     

Parker Vectors for Infinite Groups

We take the first step towards establishing a theory of Parker vectors for infinite permutation groups, with an emphasis towards oligomorphic groups. We show that, on the one hand, many results for finite groups extend naturally to the infinite case (Parker’s Lemma, multiplicative properties, etc.), while on the other, in the infinite case some genuinely new phenomena arise. We also note that calculating Parker vectors of oligomorphic groups is akin to counting circulant combinatorial objects, mirroring in a sense the combinatorial meaning of the orbit-counting sequence of an oligomorphic group. Finally we explicitly find the Parker vectors for some groups, one of which being the automorphism group of the Rado graph.     

Strongly Bounded Groups and Infinite Powers of Finite Groups

We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently initiated by Bergman.

Our main result is that G ^I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that ω₁-existentially closed groups are strongly bounded.     

Infinite Groups with an Anticentral Element

An element of a group is called anticentral if the conjugacy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camina group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camina groups, which are not locally solvable.     

Modules with Involution for Algebraic Groups

Abstract

Let k be an algebraically closed field of characteristic ≠2 and let G be a connected, reductive algebraic group with involution θ ∈ Aut(G). A (G, θ)-module is a rational G-module with a compatible action of θ. The purpose of this note is to classify the irreducible (G, θ)-modules.     

Geometrizing Infinite Dimensional Locally Compact Groups

We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.

     

Profinite Groups Admitting Just Infinite Quotients

 A profinite group is said to be just infinite if each of its proper quotients is finite. We address the question which profinite groups admit just infinite quotients. It is proved that any profinite group whose order (as a supernatural number) is divisible only by finitely many primes admits just infinite quotients. It is shown that if a profinite group G possesses the property in question then so does every open subgroup and every finite extension of G. Received 20 July 2001     

Suborbits in Infinite Primitive Permutation Groups

For every infinite cardinal , we construct a primitive permutationgroup which has a finite suborbit paired with a suborbit ofsize . This answers a question of Peter M. Neumann. 2000 MathematicsSubject Classification 20B07, 20B15, 03C50, 05C20.     

On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

We investigate the relationship between the norm N _G() of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G() is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G() is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G() has finite index in G.     

Cocompact Spherical-Euclidean Spaceform Groups of Infinite VCD

We exhibit closed manifolds M covered by S^2n–1 x R^k forall n 2 and for sufficiently large k, with fundamental groupsof infinite virtual cohomological dimension. These examplesare based on results of Raghunathan on lattices in covers ofspin and symplectic groups, and address a problem first raisedby Wall.     

关于二次模的正交群

本文给出了交换环上二次模直交和的正交群的计算公式，对多项式环上的二次模，给出了一类子群的局部整体定理。