On triple factorizations of finite groups

Triple factorizations of groups G of the form G = ABA, for proper subgroups A and B, are fundamental in the study of Lie type groups, as well as in geometry. They correspond to flag-transitive point-line incidence geometries in which each pair of points is incident with at least one line. This paper introduces and develops a general framework for studying triple factorizations of this form for finite groups, especially nondegenerate ones where G not equal AB. We identify two necessary and sufficient conditions for subgroups A, B to satisfy G = ABA, in terms of the G-actions on the A-cosets and the B-cosets. This leads to an order (upper) bound for vertical bar G vertical bar in terms of vertical bar A vertical bar and vertical bar B vertical bar which is sharp precisely for the point-line incidence geometries of flag-transitive projective planes. We study in particular the case where G acts imprimitively on the A-cosets, inducing a permutation group that is naturally embedded in a wreath product G(0) (sic) G(1). This gives rise to triple factorizations T-0, T-1, T-0 (sic) T-1 for G(0), G(1) and G(0) (sic) G(1), respectively. We present a rationale for further study of triple factorizations G = ABA in which A is maximal in G, and both A and B are core-free.