A characterization of the full wreath product

Abstract

A groupoid [3, 17] is a set Q endowed with a binary product, that is, a map from Q × Q to Q. In his 1964 paper [7], Bernd Fischer studied distributive quasigroups, which by definition are groupoids Q for which right multiplication by any fixed element gives an automorphism of Q as does left multiplication. Fischer proved that the right multiplication group R(Q) of a finite distributive quasigroup Q is solvable. He did this by showing that, for a minimal counterexample, the right multiplications T = {μa : g 7→ ga | a ∈ Q } are a generating conjugacy class of involutions in R(Q) ≤ Aut(Q) with the additional property that |tr| = 3 for distinct t and r from T . He then proved that this property forces finite R(Q) to have a normal 3-group of index 2. This led Fischer to consider [8, 9, 10] the extent to which finite symmetric groups can be characterized through being generated by a conjugacy class of involutions with all products of order 1, 2, or 3—a class of 3-transpositions, since the model is the transposition (2-cycle) class of Sym(Ω), the symmetric group on the set Ω. In a landmark theorem [10], Fischer found all finite 3transposition groups with no nontrivial solvable normal subgroups, discovering three new sporadic simple groups along the way. At the same time that Fischer was considering distributive quasigroups, George Glauberman was working on certain special groupoids, called Bruck loops. Glauberman [13] proved that finite Bruck and finite Moufang loops of odd order are solvable. His approach was similar to Fischer’s. He constructed a canonical conjugacy class T of involutory loop permutations with the additional property that |tr| was always odd for t and r from T . In his famous Z∗-theorem [14], Glauberman then proved that a finite group generated by such a class T has a normal subgroup of odd order and index 2 (a result also proved by Fischer [7] in the special case where all orders |tr| are powers of some fixed odd prime). Fischer’s and Glauberman’s work on finite quasigroups and loops had a profound effect on the theory of finite simple groups. For a normal set of involutions