11 - The classification of 3-transposition groups with trivial center

Summary

INTRODUCTION

A conjugacy class D of 3-transpositions in the group G is a class of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. Such groups were introduced and studied by Bernd Fischer who classified all finite 3-transposition groups with no nontrivial normal, solvable subgroups. His work was of great importance in the classification of finite simple groups.

The basic example of a class of 3-transpositions is the class of transpositions in any symmetric group. This was the only class which Fischer originally considered, but Roger Carter pointed out that examples could be found in several of the classical groups as well. The transvections of symplectic groups over GF(2) form a class of 3-transpositions, so additionally any subgroup of the symplectic group generated by a class of transvections is also a 3-transposition group. The symmetric groups arise in this way as do the orthogonal groups over GF(2). Symplectic transvections over GF(2) are special cases of unitary transvections over GF(4), and this unitary class is still a class of 3-transpositions. The final classical examples are given by the reflection classes of orthogonal groups over GF(3).