Book contents
- Frontmatter
- Contents
- Introduction
- Addresses of registered participants
- Addresses of non-participating authors
- Programme of lectures
- Conference photograph and key
- Symmetric presentations and orthogonal groups
- A constructive recognition algorithm for the special linear group
- Relations in M666
- A survey of symmetric generation of sporadic simple groups
- Harish-Chandra theory, q-Schur algebras, and decomposition matrices for finite classical groups
- The Meataxe as a tool in computational group theory
- Branching rules for modular projective representations of the symmetric groups
- Characters and surfaces: a survey
- On the characterization of finite groups by characters
- Finite linear groups of small degree
- Minimal parabolic systems for the symmetric and alternating groups
- Probabilistic methods in the generation of finite simple groups
- Condensing tensor product modules
- Intersections of Sylow subgroups in finite groups
- Anatomy of the Monster: I
- An integral ‘Meat-axe’
- Finite rational matrix groups: a survey
- Chamber graphs of sporadic group geometries
- An Atlas of sporadic group representations
- Presentations of reductive Fischer groups
- A brief history of the ATLAS
Probabilistic methods in the generation of finite simple groups
Published online by Cambridge University Press: 19 May 2010
- Frontmatter
- Contents
- Introduction
- Addresses of registered participants
- Addresses of non-participating authors
- Programme of lectures
- Conference photograph and key
- Symmetric presentations and orthogonal groups
- A constructive recognition algorithm for the special linear group
- Relations in M666
- A survey of symmetric generation of sporadic simple groups
- Harish-Chandra theory, q-Schur algebras, and decomposition matrices for finite classical groups
- The Meataxe as a tool in computational group theory
- Branching rules for modular projective representations of the symmetric groups
- Characters and surfaces: a survey
- On the characterization of finite groups by characters
- Finite linear groups of small degree
- Minimal parabolic systems for the symmetric and alternating groups
- Probabilistic methods in the generation of finite simple groups
- Condensing tensor product modules
- Intersections of Sylow subgroups in finite groups
- Anatomy of the Monster: I
- An integral ‘Meat-axe’
- Finite rational matrix groups: a survey
- Chamber graphs of sporadic group geometries
- An Atlas of sporadic group representations
- Presentations of reductive Fischer groups
- A brief history of the ATLAS
Summary
Abstract
We survey recent progress, made using probabilistic methods, on several problems concerning generation of finite simple groups. For example, we outline a proof that all but finitely many classical groups different from PSp4{q) (q = 2a or 3a) can be generated by an involution and an element of order 3.
Results
In this survey we present some new methods and results in the study of generating sets for the finite (nonabelian) simple groups. The results are largely taken from the three papers [16], [17], [18]. We shall present the results in this first section, and outline some proofs in sections 2 and 3. We begin by describing some of the basic questions and work in the area.
It is a well known consequence of the classification that every finite simple group can be generated by two elements. This result was established early this century for the alternating groups by Miller [23] and for the groups PSL2(q) by Dickson [8]. Various other simple groups were handled by Brahana [5] and by Albert and Thompson [1], but it was not until 1962 that Steinberg [26] showed that all finite simple groups of Lie type can be generated by two elements. To complete the picture, in 1984 Aschbacher and Guralnick [2] established the same conclusion for sporadic groups.
A refinement of the two element generation question asks whether every finite simple group can be generated by an involution and a further element. Partial results on this question were obtained in the above-mentioned papers [23], [5], [1], [2], but only recently has the question been answered completely, in the affirmative, by Malle, Saxl and Weigel [22].
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- The Atlas of Finite Groups - Ten Years On , pp. 163 - 173Publisher: Cambridge University PressPrint publication year: 1998