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The subgroup structure of finite classical groups in terms of geometric configurations

There are four classes of Classical Group. Perhaps it is most straightforward to name a significant group within each class and to then describe the various related groups in the class. Four significant groups, then, are GLn(q), Spn(q), On(q) and Un(q). We begin by describing these groups in some detail. This paper is a survey of progress towards a geometric description of the subgroup structure of the classical groups. We shall describe Aschbacher’s Theorem in some detail, even though Aschbacher’s approach is clearly not geometrical, for Aschbacher’s Theorem demonstrates very largely the structure that one should expect to find. It is unrealistic to expect to furnish a proof of Aschbacher’s Theorem that is entirely geometric, but it is a reasonable hope that nearly all the maximal subgroups of classical groups can be described geometrically and that maximality can be proven using geometric means. Maximal here means nothing more than maximal relative to inclusion. The general linear group, GLn(q), consists of the invertible linear transformations of an n-dimensional vector space V = V (n, q) over the finite field GF (q), where q = pe with p prime. Above this sits ΓLn(q), the group of all invertible semi-linear transformations of V . Below GLn(q) sits SLn(q), the set of all linear transformations of V having determinant 1; SLn(q) is a characteristic subgroup of ΓLn(q). The centre of ΓLn(q) is the group Z of non-zero scalar linear transformations; this also the centre of GLn(q). The quotients PΓLn(q) = ΓLn(q)/Z, PGLn(q) = GLn(q)/Z and PSLn(q) = Z.SLn(q)/Z act naturally on the projective space PG(n− 1, q). In this context, PΓLn(q) may be viewed as the set of all collineations of PG(n− 1, q). For some people, this is the natural geometric group. For others, PGLn(q) is the natural geometric group because its elements can be represented as matrices. From another perspective, PSLn(q) is sometimes regarded as the most important group in the class because it is usually simple. The symplectic group, Spn(q), is the group of all elements of GLn(q) preserving a non-degenerate alternating form; the non-degeneracy leads to n being even. There are various ways to describe Spn(q) in terms of matrices. One is to consider the matrix JS = [ 0 I −I 0 ] . Then Spn(q) consists of the matrices A such that AJSA = JS . Above Spn(q) sit ΓSpn(q) consisting of the invertible semi-linear transformations that preserve the alternating form up to a scalar, and GSpn(q) = ΓSpn(q)∩GLn(q). A feature of Spn(q) is that its elements already have determinant 1 so it is a subgroup of SLn(q). Each group has a corresponding projective group, namely its image in PΓLn(q). In geometric terms, PΓSpn(q) preserves a symplectic polarity. The geometric objects are the absolute subspaces with respect to this polarity, although PΓSpn(q) may be viewed as the group of collineations preserving the set of absolute lines. The group PSpn(q) is usually simple. The unitary group, Un(q), is defined when q is a square and is the group of all