3-Transposition Groups

Contains the first complete published proof of Fischer's Theorem on the classification of 3-transposition groups.

Michael Aschbacher (Author)

9780521571968, Cambridge University Press

Hardback, published 28 November 1996

272 pages, 3 b/w illus. 3 tables
23.6 x 16 x 2.5 cm, 0.515 kg

'If you have any interest in group theory, then you got to have this book.' Bulletin of the Belgian Mathematical Society

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of 3-transposition groups has become an important part of finite simple group theory, but Fischer's work has remained unpublished. 3-Transposition Groups contains the first published proof of Fischer's Theorem, written out completely in one place.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. Thus Part I 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 and are a step in the author's program to supply a strong foundation for the theory of sporadic groups.

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.

Subject Areas: Groups & group theory [PBG]