3-Transposition Groups

3-Transposition Groups

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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of 3-transposition groups has become an important part of finite simple group theory, but Fischer's work has remained unpublished. 3-Transposition Groups contains the first published proof of Fischer's Theorem, written out completely in one place.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.show more

Product details

  • Electronic book text
  • Cambridge University Press (Virtual Publishing)
  • Cambridge, United Kingdom
  • 3 b/w illus. 3 tables
  • 1139243853
  • 9781139243858

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.show more

Review quote

'Just to make available the Fischer theory to the mathematical community would justify this book. Moreover, we find a great many of interesting properties of the Fischer groups and others which are spread over the literature or may be known to specialists but cannot be found anywhere.' W. Willems, Magdeburg ' ... this book offers a profound insight into the theory of finite simple groups even for non-specialists.' G. Kowol, Monatshefte fur Mathematik 'The author's main objective is to provide a proof of the theorem classifying the almost simple groups generated by 3-transpositions, which Bernd Fischer found in 1970 but never published ... A specialised but important topic, which is given a very readable exposition here.' Mathematika 'I recommend the book strongly to two types of potential reader: the reader who wishes to see a proof of a beautiful and key theorem in the classification theorem explained by a master and the reader who is already expert in finite group theory and who wishes to gain detailed insight into the current programme of placing the theory of sporadic simple groups in a conceptual framework.' Proceedings of the Edinburgh Mathematical Society ' ... its place as an important reference work is assured, and it should be in every serious finite group theorist's library.' Bull. of London Math. Soc. 'If you have any interest in group theory, then you got to have this book.' Bulletin of the Belgian Mathematical Societyshow more