3-Transposition Groups

by Michael Aschbacher

Overview

In 1970 Bernd Fischer proved his beautiful theorem classifying the almost simple groups generated by 3-transpositions, and in the process discovered three new sporadic groups, now known as the Fischer groups. Since then, the theory of 3-transposition groups has become an important part of finite simple group theory, but Fischer's work has remained unpublished. 3-Transposition Groups contains the first published proof of Fischer's Theorem, written out completely in one place. Fischer's result, while important and deep (covering a number of complex examples), can be understood by any student with some knowledge of elementary group theory and finite geometry. Part I of this book has minimal prerequisites and could be used as a text for an intermediate level graduate course; parts II and III are aimed at specialists in finite groups.

Product Details

ISBN-13:	9780521101028
Publisher:	Cambridge University Press
Publication date:	02/05/2009
Series:	Cambridge Tracts in Mathematics , #124
Pages:	272
Product dimensions:	6.00(w) x 9.00(h) x 0.80(d)

Table of Contents

Part I. Fischer's Theorem: 1. Preliminaries; 2. Commuting graphs of groups; 3. The structure of 3-transposition groups; 4. Classical groups generated by 3-transpositions; 5. Fischer's theorem; 6. The geometry of 3-transposition groups; Part II. Existence and Uniquenesss Of The Fischer Groups: 7. Some group extensions; 8. Almost 3-transposition groups; 9. Uniqueness systems and coverings of graphs; 10. U4 (3) as a subgroup of U6 (2); 11. The existence and uniqueness of the Fischer groups; Part III. The Local Structure Of The Fischer Groups: 12. The 2-local structure of the Fischer groups; 13. Elements of order 3 in orthogonal groups over GF(3); 14. Odd locals in Fischer groups; 15. Normalisers of subgroups of prime order in Fischer groups.

Editorial Reviews

"...Aschbacher's excellent book is required for all with specific interest in finite simple groups, but those with a less direct connection will also find much of value. In particular, they will filnd the only available collected treatment of Fischer's classification of 3-transposition groups, one of the most important and historic results from the theory of finite simple groups, presented lucidly by one of the most original minds in that area." Bulletin of the American Mathematical Society

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