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Finite Permutation Groups and Finite Simple Groups

Finite Permutation Groups and Finite Simple Groups FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS PETER J. CAMERON 1. Introduction In the past two decades, there have been far-reaching developments in the problem of determining all finite non-abelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of course, the solution will have a considerable effect on many related areas, both within group theory and outside. The purpose of this article is to consider the theory of finite permutation groups with the assumption that the finite simple groups are known, and to examine questions such as: which problems are solved or solvable under this assumption, and what important problems remain? Let us begin with an example. The best-known problem in finite permutation group theory is that of deciding whether there are any 6-transitive groups other than the symmetric and alternating groups. It was conjectured by Schreier that the outer automorphism group of a finite simple group is soluble. Such a conjecture is easily checked if the list of simple groups is known. (Indeed, at present it seems very likely that Schreier's conjecture will http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Finite Permutation Groups and Finite Simple Groups

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References (63)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/13.1.1
Publisher site
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Abstract

FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS PETER J. CAMERON 1. Introduction In the past two decades, there have been far-reaching developments in the problem of determining all finite non-abelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of course, the solution will have a considerable effect on many related areas, both within group theory and outside. The purpose of this article is to consider the theory of finite permutation groups with the assumption that the finite simple groups are known, and to examine questions such as: which problems are solved or solvable under this assumption, and what important problems remain? Let us begin with an example. The best-known problem in finite permutation group theory is that of deciding whether there are any 6-transitive groups other than the symmetric and alternating groups. It was conjectured by Schreier that the outer automorphism group of a finite simple group is soluble. Such a conjecture is easily checked if the list of simple groups is known. (Indeed, at present it seems very likely that Schreier's conjecture will

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1981

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