Synopses & Reviews
The finite simple groups are the building blocks from which all the finite groups are made and as such are objects of fundamental importance throughout mathematics. They can be divided into the alternating, sporadic and Lie type groups, the latter further subdivided into classical and exceptional, or into untwisted and twisted types. This book is the first accessible introduction to all these families of finite simple groups at a level suitable for final year undergraduate and beginning graduate students.
The first five chapters provide a thorough grounding in the theory of the alternating and classical groups, followed by an overview of the exceptional groups (treated as automorphism groups of multilinear forms) and the sporadic groups. These chapters form the basis of a final year undergraduate course bringing their undergraduate studies to a fitting climax with seminal results from the late 20th century.
The final two chapters give an introduction to the theory of Lie algebras and Chevalley groups (which provides a unified approach to all the untwisted finite groups of Lie type) and to algebraic groups (which unites the twisted and untwisted types). These final chapters are ideal guides for undergraduate projects and prepare the students for further reading in more advanced texts on these important topics.
Review
From the reviews: "The book under review has as its main goal to give an introductory overview of the construction and main properties of all finite simple groups. ... This book is the first one that attempts to give a systematic treatment of all finite simple groups, using the more recent and efficient constructions ... . The author succeeds in making this important but difficult area of mathematics readily accessible to a large sector of the mathematical community, and for this we should be grateful." (Felipe Zaldivar, The Mathematical Association of America, March, 2010)
Synopsis
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].
Synopsis
The finite simple groups are the building blocks from which all the finite groups are made and as such they are objects of fundamental importance throughout mathematics. The classification of the finite simple groups was one of the great mathematical achievements of the twentieth century, yet these groups remain difficult to study which hinders applications of the classification. This textbook brings the finite simple groups to life by giving concrete constructions of most of them, sufficient to illuminate their structure and permit real calculations both in the groups themselves and in the underlying geometrical or algebraic structures. This is the first time that all the finite simple groups have been treated together in this way and the book points out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and discusses a number of new approaches to some of the groups. Many exercises of varying difficulty are provided. The Finite Simple Groups is aimed at advanced undergraduate and graduate students in algebra as well as professional mathematicians and scientists who use groups and want to apply the knowledge which the classification has given us. The main prerequisite is an undergraduate course in group theory up to the level of Sylow's theorems.
Synopsis
Here, a thorough grounding in the theory of alternating and classical groups is followed by discussion of exceptional groups (classed as automorphism groups of multilinear forms), sporadic and Chevalley groups, as well as the theory of Lie algebras.
Table of Contents
Introduction.- The alternating groups.- The classical groups.- The exceptional groups.- The sporadic groups.