The general theory of 3-transposition groups

Abstract

A set D of 3-transpositions in the group G is a normal set 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. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.