Finite Groups that Admit Kantor Families

The basic problem of interest in this note is: What finite groups G admit a Kantor family (i.e., 4-gonal family) of subgroups? We begin with a survey of the known examples (of the groups G, not of the Kantor families) and then pose the question of whether or not two specific examples are isomorphic. The two groups in question have order q and coexist for q = 3 ≥ 27. Our conjecture had been that they are not isomorphic. During the conference it was announced that indeed they are not isomorphic, but the proof will be published elsewhere. 1. What is a Kantor Family? In 1980 W. M. Kantor [Ka80] found a new recipe for the construction of finite generalized quadrangles (GQ). The chief ingredient of this recipe is a family of subgroups of a given group satisfying a pair of conditions that are readily translated into necessary and sufficient conditions for a certain coset geometry to be a finite GQ. In this note we are not concerned with the GQ per se, but only with the groups that admit these Kantor families, which we now describe. Let s and t be positive integers greater than 1, and let G be a group of order st. Let F be a family F = {Ai : 0 ≤ i ≤ t} of t+ 1 subgroups of G of order s for which the following condition is satisfied: K1. Ai ·Aj ∩Ak = {id} whenever i, j, k are distinct, 0 ≤ i, j, k ≤ t. Put Ω = ∪(Ai : 0 ≤ i ≤ t). Then for each i, 0 ≤ i ≤ t define a subset Ai of G by Ai = Ai ∪ (Aig : g ∈ G and Aig ∩ Ω = ∅)). Put F∗ = {Ai : 0 ≤ i ≤ t}. It follows that |Ai | = st for each i. If each Ai is actually a subgroup of G, then Ai ≤ Ai for all i and the following second condition of Kantor is also satisfied: K2. Ai ∩Aj = {id} whenever i 6= j, 0 ≤ i, j ≤ t.