Pairwise generating and covering sporadic simple groups.
type: Item
instance of: scholarly article
title: Pairwise generating and covering sporadic simple groups.
zbMATH Open document ID: 1202.20019
author: Petra E. Holmes
author: Attila Maróti
published in: Journal of Algebra
publication date: 2010-07-20T00:00:00Z
review text: Let \(G\) be a non-cyclic finite group that can be generated by two elements. A subset \(S\) of \(G\) is said to be a pairwise generating set for \(G\) if every distinct pair of elements in \(S\) generates \(G\). The maximal size of a pairwise generating set for \(G\) is denoted by \(\omega(G)\). The minimal number of proper subgroups of \(G\) whose union is \(G\) is denoted by \(\sigma(G)\). Obviously, \(\omega(G)\leq\sigma(G)\) if \(G\) is non-cyclic. In the recent years the precise values or good approximations of \(\omega(G)\) and \(\sigma(G)\) have been obtained for some relevant families of finite simple groups. The methods so far developed are efficient only when the order of \(G\) is ``large''. It should be interesting to develop techniques that work for ``small'' finite simple groups. In this paper the authors provide lower bounds for \(\omega(G)\) and upper bounds for \(\sigma(G)\) whenever \(G\) is one of the 26 sporadic simple groups. In particular the exact value of \(\sigma(G)\) is determined for \(G\) any of the groups \(M_{11}\), \(M_{22}\), \(M_{23}\), \(M_{24}\), \(Fi_{22}\), \(He\), \(Ru\), \(HS\), \(HN\), \(O'N\) and \(Ly\).
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 5758105
zbMATH Keywords: sporadic simple groups
zbMATH Keywords: generation
zbMATH Keywords: subgroup coverings
zbMATH Keywords: graphs
zbMATH Keywords: pairwise generating sets
describes a project that uses: GAP
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.10.011
OpenAlex ID: W2056994767
cites work: Q5307729
cites work: Sets of permutations that generate the symmetric group pairwise.
cites work: Sets of elements that pairwise generate a linear group
cites work: On $n$-Sum Groups.
cites work: Q3684278
cites work: Q5505822
cites work: The probability of generating the symmetric group
cites work: Subgroup coverings of some sporadic groups.
cites work: The probability of generating a finite classical group
cites work: The probability of generating a finite simple group
cites work: Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky
cites work: Groups Covered By Permutable Subsets
cites work: Groups as the union of proper subgroups.
cites work: Q5781249
DOI: 10.1016/J.JALGEBRA.2009.10.011
description: scientific article; zbMATH DE number 5758105