\(2\)-generations of the fourth Janko group \(J_4\)
type: Item
instance of: scholarly article
title: \(2\)-generations of the fourth Janko group \(J_4\)
zbMATH Open document ID: 0918.20022
author: Shahiem Ganief
author: Jamshid Moori
published in: Journal of Algebra
publication date: 1999-08-16T00:00:00Z
review text: This paper is devoted to a detailed study of the different ways in which the group of the title can be generated by two elements, \(a\) and \(b\), say. We add a redundant generator \(c=b^{-1}a^{-1}\), and by symmetry we may assume that \(o(a)\leq o(b)\leq o(c)\). If these three orders are \(l\), \(m\), \(n\), we speak of \((l,m,n)\)-generation, etc., and if \(a\), \(b\), \(c\) are specified further to lie in particular conjugacy classes \(lX\), \(mY\), \(nZ\), we speak of \((lX,mY,nZ)\)-generation. Sections 2-4 are devoted to determining precisely which such triples of conjugacy classes arise for generating pairs as above, in the Janko group \(J_4\). The authors use the completeness of the list of maximal subgroups in an essential way. The second question is, for each conjugacy class \(C\): can the group \(J_4\) be generated by an element in \(C\) and an element \(y\) in any non-trivial conjugacy class \(Y\) whatsoever? This is known as \(C\)-complementary generation, and is relatively easy to determine, once the answer to the first question is known. Several earlier papers by the same authors, performing the same calculations in other sporadic simple groups, are referred to.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 1271655
zbMATH Keywords: conjugacy classes
zbMATH Keywords: generating pairs
zbMATH Keywords: Janko group \(J_4\)
zbMATH Keywords: maximal subgroups
zbMATH Keywords: \(C\)-complementary generations
reviewed by: Robert A. Wilson
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: jabr.1998.7578
OpenAlex ID: W2046308713
cites work: Some results on quotients of triangle groups
cites work: The Symmetric Genus of Sporadic Groups
cites work: Q3684278
cites work: Q3973544
cites work: (2,3,t)-Generations for the janko group j3
cites work: rX-Complementary generations of the janko groups j1J2And j3
cites work: \((p,q,r)\)-generations and \(nX\)-complementary generations of the sporadic groups \(HS\) and \(McL\)
cites work: \((p,q,r)\)-generations of the smallest Conway group \(Co_ 3\)
cites work: Q4347287
cites work: Generating pairs for the Conway groups Co2 and Co3
cites work: A new finite simple group of order 86,775,571,046,077,562,880 which possesses \(M_{24}\) and the full covering group of \(M_{22}\) as subgroups
cites work: Q3831183
cites work: The Maximal Subgroups of J 4
cites work: Q3129418
cites work: (2,3,p)-generations for the fischer group f22
cites work: Q3894104
cites work: Matrices and cohomology
cites work: Representing M11, M12, M22, and M23on surfaces of least genus
cites work: Sporadic simple groups which are Hurwitz
cites work: 3/2—Generation of the sporadic simple groups
DOI: 10.1006/JABR.1998.7578
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description: scientific article; zbMATH DE number 1271655