Classical groups, probabilistic methods, and the \((2,3)\)-generation problem
type: Item
instance of: scholarly article
title: Classical groups, probabilistic methods, and the \((2,3)\)-generation problem
zbMATH Open document ID: 0865.20020
DOI: 10.2307/2118584
author: Martin W. Liebeck
author: Aner Shalev
published in: Annals of Mathematics. Second Series
publication date: 1996-11-27T00:00:00Z
review text: A group is said to be \((m,n)\)-generated if it is generated by a pair of elements \(x\), \(y\), say, where \(x\) (resp. \(y\)) has order \(m\) (resp. \(n\)). It is known (assuming the classification of finite simple groups) that every such group is generated by a pair of elements. The natural question then arises: to what extent can the orders of such a pair of generators be restricted? This has led to the following: Conjecture. With some exceptions of low rank in characteristic 2 and 3, every finite (non-abelian) simple group is \((2,3)\)-generated. This represents the ``best possible'' situation since every \((2,2)\)-generated group is metabelian. The conjecture implies that almost all finite simple groups are images of the modular group \(\text{PSL}_2(\mathbb{Z})\). Using deterministic methods (which (more or less) explicitly provide generators), a number of authors have proved that many finite simple groups are \((2,3)\)-generated. This paper is to date the most important contribution in support of the above conjecture. The authors use a probabilistic approach which is originally due to Dixon (1969). Let \(G\) be a finite simple classical (or alternating) group. Let \(X\) be the set of all ordered pairs \((x,y)\) in \(G\times G\) where \(x\) (resp. \(y\)) has order 2 (resp. 3) and let \(Y\) be the subset of \(X\) consisting of all such pairs which generate \(G\). Then \(P_{2,3}(G)=|Y|/|X|\) measures the probability that a randomly chosen pair of elements of orders 2 and 3 generate \(G\). The complementary probability \(Q_{2,3}(G)=1-P_{2,3}(G)\) involves those pairs in \(X\) which lie in some maximal subgroup of \(G\). Using the Classification together with many important results on maximal subgroups, the authors prove that \(Q_{2,3}(G)\) is bounded above (via an absolute constant) by the values of a ``zeta function'' \(\zeta_G(s)\), for some fixed \(s\). The function \(\zeta_G(s)\), which has only finitely many terms, encodes the indices of the maximal subgroup…
reviewed by: A. W. Mason
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 935302
zbMATH Keywords: \((2,3)\)-generated finite simple groups
zbMATH Keywords: pairs of generators
zbMATH Keywords: finite simple groups
zbMATH Keywords: modular group
zbMATH Keywords: randomly chosen pairs of elements
zbMATH Keywords: maximal subgroups
zbMATH Keywords: finite simple classical groups
zbMATH Keywords: conjugate involutions
Wikidata QID: Q59541318
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118584
OpenAlex ID: W2323383486
description: scientific article; zbMATH DE number 935302