By Michael Aschbacher
Cambridge Tracts in Mathematics (No. 124)
Publisher: Cambridge University Press
Print Publication Year: 1996
Online Publication Date:August 2010
Online ISBN:9780511759413
Hardback ISBN:9780521571968
Paperback ISBN:9780521101028
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511759413.007
Subjects: Algebra
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We have already made extensive use of finite geometry in our study of groups generated by 3-transpositions. In this chapter we consider another type of geometry associated with a 3-transposition group G. This geometry is the Fischer space of G. The notion is due to F. Buekenhout [Bu].
Fischer spaces are a special class of partial linear spaces. Steiner triple systems are also partial linear spaces, and the Fischer space of a 3-transposition group of width 1 is a triple system. In Section 19 we prove a result of M. Hall [H4] that can be interpreted as classifying all 4-generator 3-transposition groups of width 1, or as classifying certain Steiner triple systems. Recall we used Hall's result to prove Lemma 8.6.
Fischer's Theorem gives a complete description of finite almost simple 3-transposition groups. Section 20 consists of a survey of results in the literature on 3-transposition groups that are not finite or not almost simple. Much of this work uses Fischer spaces, so Section 20 contains further discussion of those objects. Section 20 also contains some discussion of generalizations of the 3-transposition condition. In particular we find that the long root elements of groups of Lie type are naturally described by one such generalization.
pp. i-iv
pp. v-viii
Part I - Fischer's Theory
pp. 1-5
pp. 6-20
2 - Commuting graphs of groups: Read PDF
pp. 21-29
3 - The structure of 3-transposition groups: Read PDF
pp. 30-38
4 - Classical groups generated by 3-transpositions: Read PDF
pp. 39-57
5 - Fischer's Theorem: Read PDF
pp. 58-90
6 - The geometry of 3-transposition groups: Read PDF
pp. 91-108
Part II - The existence and uniqueness of the Fischer groups
pp. 109-112
7 - Some group extensions: Read PDF
pp. 113-136
8 - Almost 3-transposition groups: Read PDF
pp. 137-146
9 - Uniqueness systems and coverings of graphs: Read PDF
pp. 147-156
10 - U4(3) as a subgroup of U6(2): Read PDF
pp. 157-172
11 - The existence and uniqueness of the Fischer groups: Read PDF
pp. 173-198
Part III - The local structure of the Fischer groups
pp. 199-200
12 - The 2-local structure of the Fischer groups: Read PDF
pp. 201-210
13 - Elements of order 3 in orthogonal groups over GF(3): Read PDF
pp. 211-225
14 - Odd locals in Fischer groups: Read PDF
pp. 226-249
15 - Normalizers of subgroups of prime order in Fischer groups: Read PDF
pp. 250-252
pp. 253-254
pp. 255-256
pp. 257-260