6 - The geometry of 3-transposition groups

Summary

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.