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
A survey of symmetric generation of sporadic 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
Many of the sporadic simple groups possess highly symmetric generating sets which can often be used to construct the groups, and which carry much information about their subgroup structure. We give a survey of results obtained so far.
Introduction and motivation
This paper is concerned with groups which are generated by highly symmetric subsets of their elements: that is to say by subsets of elements whose set normalizer within the group they generate acts on them by conjugation in a highly symmetric manner. Rather than investigate the behaviour of various known groups, we turn the procedure around and ask what groups can be generated by a set of elements which possesses certain assigned symmetries. It turns out that this approach enables us to define and construct by hand a large number of interesting groups—including many of the sporadic simple groups.
Accordingly we let m*n denote Cm*Cm* – *Cm, a free product of n copies of the cyclic group of order m. Let F = T0*T1* … *Tn−1 be such a group, with Ti = 〈ti〉 ≅ Cm. Certainly permutations of the set of symmetric generators T = {t0, t1, …, tn−1} induce automorphisms of F. Further automorphisms are given by raising a given ti to a power of itself coprime to m, while fixing the other symmetric generators. Together these generate the group M of monomial automorphisms of F which is a wreath product HrSn, where Hr is an abelian group of order r = Φ(m), the number of positive integers less than m and coprime to it.
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- Information
- The Atlas of Finite Groups - Ten Years On , pp. 39 - 57Publisher: Cambridge University PressPrint publication year: 1998
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