On triple factorisations of finite groups

Factorising is a general method for reducing the complexity of a problem and elucidating its structure. This is true not only for factorising numbers or matrices but also for factorising algebraic structures such as groups as products of proper subgroups. For example, group factorisations G = AB, where A and B are proper subgroups, have been extensively studied. Many of these results determine various group theoretic properties of G from properties of the subgroups A and B. We propose a general study of a larger class of group factorisations, namely, G = ABA, where A and B are proper subgroups of G. In this case, the triple T = (G; A;B) is said to be a triple factorisation of G. The triple factorisations are connected with permutation group theory as well as geometry, abstract group theory and the theory of Lie type groups and algebras; however, up to now it has been developed more in the context of geometry, abstract roup theory and Lie type group theory. In this thesis, we intend to study triple factorisations of groups using permutation group theory.

Triple factorisations arise naturally from groups with BN-pairs. The most well known example arises for general linear groups G = GL(n; F) of non-singular nxn matrices over a field F; for B the subgroup of upper triangular matrices and N the subgroup of monomial matrices we have the triple factorisation G = BNB. In fact, each group G with a BN-pair has a triple factorisation T = (G;B;N), and we know that `Chevalley groups' and `twisted groups' have BN-pairs. The importance of these groups, both in group theory and representation theory is one of the substantial reasons behind this thesis to study triple factorisations of groups. Another part of our interest stems from the fact that each triple factorisation G = ABA corresponds to a rank 2 Buekenhout geometry with points the A-cosets and lines the B-cosets, and incidence given by nontrivial intersection; the group G acts fag-transitively, and for each pair of points, there is at least one line incident with both of them. Moreover, it is geometrically shown that such a group acts primitively on the points, and this give rise a substantial reason to to study flag-transitive primitive groups on the points. Further examples of factorisations G =ABA, with G almost simple, include all known (3/2)-transitive almost simple groups.

We also have shown that the triple factorisation G = ABA is equivalent to the subset Gamma:= {Ba | a in G} having restricted movement under the multiplication action of G on right cosets of B, that is, each Gamma^g:={Bag | a in A} nontrivially intersects Gamma. Thus, the theory of triple factorisations is intimately linked with that of subsets of restricted movement going back to the `SeparationTheorem' of B. H. Neumann in 1954, and including Peter Neumann's theory of close-knit families of sets, and work of Brailovsky, Pasechnik and Praeger on quasi-invariant sets. Therefore, research about triple factorisations T = (G, A,B), especially in the case where G is an almost simple group, will impact on the study of fag-transitive designs and on the separation properties for permutation groups.

In addition to the links mentioned above between triple factorisations, and geometry, and separation properties of permutation groups, several special kinds of triple factorisations have been studied in a purely group-theoretic context. The finite triple factorisations T = (G, A,B), where A, B are cyclic subgroups, were initially investigated by D. Gorenstein and I. N. Herstein in 1959. They proved that if A and B are cyclic of relatively prime orders h and k, then G is soluble, its Sylow subgroups are abelian or quaternion, and |G| = hkw, where w divides some power of k. Since then studies have been made of other conditions on A and B which imply that G is soluble. For example, M. Guterman (1969) proved the solubility of the group G when B is abelian of odd order, or B is nilpotent and A is self-normalizing. It has also been proved that if A is abelian and B is cyclic, or if A is a solvable TI-subgroup and B is a cyclic group with (|A|; |B|) = 1, then G is solvable. On the other hand, when G is a fnite nonabelian simple group, and both A and B are abelian, D. L. Zagorin and L. S. Kazarin (1996) showed that G is isomorphic to SL(2; 2^n) for appropriate n, and |A| = 2^n + 1, and |B| = 2^n.