The probability of generating a finite linear group

We study the asymptotic behavior of the probability of generating a finite completely reducible linear group G of degree n with [ n] elements. In particular we prove that if 3/2 and n is large enough then [ n] randomly chosen elements that generate G modulo O²(G) almost certainly generate G itself.Received: 13 February 2003     

The probability of generating a permutation group

It is well known that a permutation group of degree can be generated by elements. In this paper we study the asymptotic behavior of the probability of generating a permutation group of degree n with elements. In particular we prove that if n is large enough and elements generate a permutation group G of degree n modulo G G ², then almost certainly these elements generate G itself. Received: 2 January 2002     

The probability of generating a finite simple group

We study the probability of generating a finite simple group, together with its generalisation P _G,socG (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/socG. We prove that P _G,socG (2) ? 53/90, with equality if and only if G is A₆ or S₆, and establish a similar result for P _G,socG (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.     

Computation of simple characters of a Chevalley group

The purpose of this paper is to describe an algorithm for computing weight multiplicities in a simple module of an algebraic Chevalley group over a field of positive characteristicp. The method is essentially more efficient than the one introduced by N. Burgoyne in 1971, especially whenp is small. A concrete application is outlined whereG is of typeG ₂ andp=5.     

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A$A$ and B$B$ are nontrivial conjugacy classes, then AB$AB$ is not a conjugacy class. We also prove that if G$G$ is a finite simple group of Lie type and A$A$ and B$B$ are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB$AB$ is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for p$p$-elements with p≥5$p\ge 5$.     

The permutation class group of a finite group

Analogously to the projective class group, the permutation class group of a finite group π can be defined as the group of equivalence classes of direct summands of integral permutation modules modulo permutation modules. It is shown that this group behaves nicely with respect to localization and completion, which then is used to prove that contrary to the projective class group - it is not always a torsion group. More precisely, the rank of the permutation class of group is computed.     

A description of Baer-Suzuki type of the solvable radical of a finite group

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g∈G such that for any 3 elements a₁,a₂,a₃∈G the subgroup generated by the elements , i=1,2,3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}^′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).     

The probability of generating a finite classical group

In this paper generalized Fischer spaces are defined and some results on their classification are obtained.     

The Divisibility Graph of finite groups of Lie type

The Divisibility Graph of a finite group G has vertex set the set of conjugacy class sizes of non-central elements in G and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic.     

The probability of generating a nilpotent subgroup of a finite group

It is proved that for finite groups G the probability that tworandomly chosen elements of G generate a nilpotent subgrouptends to 0 as the index of the Fitting subgroup of G tends toinfinity.     

The probability of generating a soluble subgroup of a finite group

It is proved that for finite groups G, the probability thattwo randomly chosen elements of G generate a soluble subgrouptends to zero as the index of the largest soluble normal subgroupof G tends to infinity.     

Application of a Tauberian theorem to finite model theory

An extension of a Tauberian theorem of Hardy and Littlewood is proved. It is used to show that, for classes of finite models satisfying certain combinatorial and growth properties, Cesàro probabilities (limits of average probabilities over second order sentences) exist. Examples of such classes include the class of unary functions and the class of partial unary functions. It is conjectured that the result holds for the usual notion of asymptotic probability as well as Cesàro probability. Evidence in support of the conjecture is presented.     

Abelian groups with a vanishing homology group

An abelian group A is called absolutely abelian, if in every central extension N ? G ? A the group G is also abelian. The abelian group A is absolutely abelian precisely when the Schur multiplicator H₂A vanished. These groups, and more generally groups with H_nA = 0 for some n, are characterized by elementary internal properties. (Here H₁A denotes the integral homology of A.) The cases of even n and odd n behave strikingly different. There are 2^?ο different isomorphism types of abelian groups A with reduced torsion subgroup satisfying H_2nA = 0. The major tools are direct limit arguments and the Lyndon-Hochschild-Serre (L-H-S) spectral sequence, but the treatment of absolutely abelian groups does not use spectral sequences. All differentials d^r for r ≥ 2 in the L-H-S spectral sequence of a pure abelian extension vanish. Included is a proof of the folklore theorem, that homology of groups commutes with direct limits also in the group variable, and a discussion of the L-H-S spectral sequence for direct limits.     

Periodicity of Adams operations on the Green ring of a finite group

The Adams operations and on the Green ring of a group G over a field K arise from the study of the exterior powers and symmetric powers of KG-modules. When G is finite and K has prime characteristic p we show that and are periodic in n if and only if the Sylow p-subgroups of G are cyclic. In the case where G is a cyclic p-group we find the minimum periods and use recent work of Symonds to express in terms of .     

Finite traces and representations of the group of infinite matrices over a finite field

The article is devoted to the representation theory of locally compact infinite-dimensional group GLB$\mathbb{GLB}$ of almost upper-triangular infinite matrices over the finite field with q   elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate n=∞$n=\infty$ analogue of general linear groups GL(n,q)$\mathbb{GL}(n,q)$. It serves as an alternative to GL(∞,q)$\mathbb{GL}(\infty ,q)$, whose representation theory is poor.     

An explicit formula for the action of a finite group on a commutative ring

Let G be a finite group, k a commutative ring upon which G acts. For every subgroup H of G, the trace (or norm) map is defined. is onto if and only if there exists an element x_H such that . We will show that the existence of x_P for every subgroup P of prime order determines the existence of x_G by exhibiting an explicit formula for x_G in terms of the x_P, where P varies over prime order subgroups. Since is onto if and only if is, where g∈G is an arbitrary element, we need to take only one P from each conjugacy class. We will also show why a formula with less factors does not exist, and show that the existence or non-existence of some of the x_P’s (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.     

Axiomatization of abelian-by-G groups for a finite group G

We show that, for each finite group G, there exists an axiomatization of the class of abelian-by-G groups with a single sentence. In the proof, we use the definability of the subgroups M ⁿ in an abelian-by-finite group M, and the Auslander-Reiten sequences for modules over an Artin algebra. Received: 15 March 1996 / Published online: 18 July 2001     

Every group is the automorphism group of a rank-3 matroid

Every group is the automorphism group of a rank-3 extension of a rank-3 Dowling geometry.Partially supported by The George Washington University UFF grant.Partially supported by the National Security Agency under grant MDA904-91-H-0030.