On the generation of finite simple groups by pairs of subgroups
type: Item
instance of: scholarly article
title: On the generation of finite simple groups by pairs of subgroups
zbMATH Open document ID: 0648.20025
DOI: 10.1016/0021-8693(88)90220-7
author: John S. Wilson
author: M. Chiara Tamburini
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let A and B be finite non-trivial groups. The authors prove that for `almost all' pairs A, B there is a finite simple group which is generated by A and B. More precisely: Theorem 1. If \(| A| | B| \geq 12\) and q is any prime power, then for all \(n\geq 4| A| | B| +12\), the group \(G=PSL_ n(q)\) has subgroups \(\bar A\cong A\) and \(\bar B\cong B\) such that \(G=<\bar A,\bar B>\). - Theorem 2. If \(| A| | B| \geq 12\), then for all \(n\geq 4| A| | B| +12\) the alternating group \(A_ n\) has subgroups \(\bar A\cong A\) and \(\bar B\cong B\) such that \(A_ n=<\bar A,\bar B>.\) These two theorems have natural corollaries for generation of \(PSL_ n(q)\) and \(A_ n\) by pairs of elements. As an illustration of their general method the authors give a short proof that if \(n\geq 7\) then every group \(PSL_ n(q)\) can be generated by an involution and an element of order n-1.
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 4057777
zbMATH Keywords: finite simple group
zbMATH Keywords: generation of \(PSL_ n(q)\) and \(A_ n\) by pairs of elements
zbMATH Keywords: involution
reviewed by: S. A. Syskin
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90220-7
OpenAlex ID: W2159524776
cites work: Two-element generation of the projective unimodular group
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description: scientific article; zbMATH DE number 4057777