Generating finite 3-transposition groups

Abstract

By a result of Cuypers and Hall, any finitely generated 3-transposition group is finite. For each finite 3-transposition group we find the minimum number of 3-transpositions needed for generation. As a consequence we classify the central types of all 3-transposition groups generated by at most six of their 3-transpositions, confirming earlier results of Fischer, Zara, Hall, and Soicher on groups generated by at most five 3-transpositions. We also prove that for every r the order of a 3-transposition group generated by r of its 3-transpositions is bounded above by an explicit function of r. In particular, the number of such r-generated 3-transposition groups is finite.