S. Hassan Alavi, Cheryl E. Praeger
October 13, 2010
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 ≠ AB . We identify two necessary and suffcient 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 | G | in terms of | A | and | B | 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 ≀ G 1 . This gives rise to triple factorizations for G 0 , G 1 and G 0 ≀ 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.