The Finite Simple Groups

verfasst von: Professor Robert A. Wilson

Verlag: Springer London

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Thisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

The study of (non-abelian) finite simple groups can be traced back at least as far as Galois, who around 1830 understood their fundamental significance as obstacles to the solution of polynomial equations by radicals (square roots, cube roots, etc.). From the very beginning, Galois realised the importance of classifying the finite simple groups, and knew that the alternating groups A_n are simple for n≥5, and he constructed (at least) the simple groups PSL₂(p) for primes p≥5.

Robert A. Wilson

2. The alternating groups

Abstract

The most familiar of the (finite non-abelian) simple groups are the alternating groups A_n, which are subgroups of index 2 in the symmetric groups S_n. In this chapter our main aims are to define these groups, prove they are simple, determine their outer automorphism groups, describe in general terms their subgroups, and construct their covering groups. At the end of the chapter we briefly introduce reflection groups as a generalisation of the symmetric groups, as they play an important role not only in the theory of groups of Lie type, but also in the construction of many sporadic groups, as well as in the elucidation of much exceptional behaviour of low-dimensional classical groups.

Robert A. Wilson

3. The classical groups

Abstract

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

Robert A. Wilson

4. The exceptional groups

Abstract

It is the aim of this chapter to describe the ten families of so-called ‘exceptional groups of Lie type’. There are three main ways to approach these groups. The first is via Lie algebras, as is wonderfully developed in Carter’s book 21. The second, more modern, approach is via algebraic groups (see for example Geck’s book ‘Introduction to algebraic geometry and algebraic groups’ [65]). The third is via ‘alternative’ algebras, as in ‘Octonions, Jordan algebras and exceptional groups’ by Springer and Veldkamp [158]. I shall adopt the ‘alternative’ approach, for a number of reasons: although it lacks the elegance and uniformity of the other approaches, it gains markedly when it comes to performing concrete calculations. We obtain not only the smallest representations in this way, but also construct the (generic) covering groups, whereas the Lie algebra approach only constructs the simple groups.

Robert A. Wilson

5. The sporadic groups

Abstract

In this chapter we introduce the 26 sporadic simple groups. These are in many ways the most interesting of the finite simple groups, but are also the most difficult to construct. It is not possible here to provide complete proofs in all cases, but merely to indicate the general lines such proofs might take. Roughly speaking, proofs are given as far as the middle of Section 5.7, which deals with the Fischer groups. Section 5.2 deals with the large Mathieu groups M₂₄, M₂₃ and M₂₂, and then the small Mathieu groups M₁₂ and M₁₁ are treated in Section 5.3. These groups have been known since Mathieu’s papers [130, 131, 132] of the 1860s and 1870s. In all these cases it is possible to give complete constructions in a reasonably small number of pages, computing the group orders and proving simplicity, as well as exhibiting a number of important subgroups. Other facts are stated with varying degrees of justification.

Robert A. Wilson

Backmatter

Titel: The Finite Simple Groups
verfasst von: Professor Robert A. Wilson
Verlag: Springer London
Electronic ISBN: 978-1-84800-988-2
Print ISBN: 978-1-84800-987-5
DOI: https://doi.org/10.1007/978-1-84800-988-2