Introduction

Summary

Around 1970 Bemd Fischer proved his beautiful theorem classifying almost simple finite groups generated by 3-transpositions, and in the process discovered three new sporadic simple groups, now termed Fischer groups. Fischer's Theorem was deep; the list of groups appearing in its conclusion included not only the three sporadic Fischer groups but also the symmetric groups and several families of classical groups. Nevertheless Fischer used none of the sophisticated machinery of the day. His proof required little more than some elementary group theory and finite geometry, combined in a new and original way.

The hypotheses of Fischer's Theorem were unusual for the time too. Fischer considered the following setup. Let G be a finite group. A set of 3-transpositions of G is a set D of involutions of G (i.e., elements of order 2) such that D is the union of conjugacy classes of G, D generates G, and for all a, b in D, the order of the product ab is 1, 2, or 3.

One example comes to mind immediately; namely the class D of transpositions forms a conjugacy class of 3-transpositions of any symmetric group G. For if t = (x, y) and s = (u, v) are distinct transpositions then ts has order 2 if {x, y} ∩ {u, v} is empty, whereas ts has order 3 if the intersection is nonempty.

Fischer then imposed some extra constraints to single out the most interesting examples, which are almost simple.