Up-scaling hydrological processes and the development of a large-scale river basin modelling system

Let Gm be the normalizer of a symplectic type r-group Rm in SLrm (q) or SUrm (q) acting on a vector space v of dimension rm; when is Gm a maximal subgroup of the group, SLrm(q) or SUrm(q) where q = pf and f is odd or even respectively? We shall attempt to reduced this problem to the singular case of whether G1 is a maximal subgroup of SLr(q) or SUr(q) respectively when r is odd, and solve the problem completely when r = 2. We will give a detailed picture of the structure of Gm and describe how the elements of Gm act by conjugation on the elements of Rm. We will describe a Sylow r subgroup of Gm and detail some of the properties of this Sylow r subgroup, which will allow us to show that Rm is a characteristic subgroup of Gm. Earlier we will show that all the elements of Rm are conjugate in Gm. This will allow us to show that if there are any proper overgroups K of Gm in SLrm(q) or SUrm(q) in the respective cases then, either Rm intersection Rmk = {I-bar} for each k is an element of K backslash Gm or for each element s is an element of Rm there exists k is an element of K backslash Gm such that sk = s. It will then be shown that this former case cannot occur when r = 2, and evidence will be given that it can not occur when r is odd. Assuming that the former case of the above cannot occur then, Gm is a maximal subgroup of the group, SLrm(q) or SUrm(q) if and only if, when r is odd, G1 is a maximal subgroup of SLr(q) or SUr(q) respectively, and when r=2, G2 is a maximal subgroup of SLr2(q) or SUr2(q). It will then be proved that G2 is a maximal subgroup of SLr2(q) to complete the theorem for the r = 2 case. (author)