Classification of finite subgroups of \(2\times 2\) matrices over a division algebra of characteristic zero (Q1114019)

In this fine and long piece of work the author completely carries out the classification of the title. The somewhat lengthly history of this problem is well summarized in the introduction and we refer the reader to that. However we should at least mention \textit{S. A. Amitsur}'s pioneering classification of the finite subgroups of division rings [Trans. Am. Math. Soc. 80, 361-386 (1955; Zbl 0065.256); alternatively see Section 2.1 of ``Skew Linear Groups'' by \textit{M. Shirvani} and the reviewer (Lond. Math. Soc. Lect. Note Ser. 118, 1986; Zbl 0602.20046)]. Amitsur's classification consists of one insoluble group and a host of types of soluble groups. The same general pattern is followed here, but not surprisingly the lists of groups are much longer and the groups more complicated to describe. If either the subgroup G of GL(2,D) is imprimitive or if \({\mathbb{Q}}[G]\) is not simple of degree 2 the structure of G can be pieced together from groups on Amitsur's list. The author therefore concentrates on those subgroups G that are primitive and satisfy \({\mathbb{Q}}[G]=D^{2\times 2}\). The first real task is the elucidation of the Fitting subgroup of G. There are six types of possibility for the Sylow 2-subgroup of the Fitting subgroup. These are used to divide the soluble case into two fundamental types (exceptionally \(Q_ 8\) appears in both types according to its automorphisms induced by G). These are then analyzed separately. Much detailed case-by-case analysis is required. The number of different basic cases depends to some extent on how you label them, but one is looking at something like 40 types of cases just in the situation where G is soluble (and primitive etc.). The insoluble list is, as expected, much shorter, consisting only of 9 basic types. The starting point is the classification of the perfect subgroups of GL(2,D) with \({\mathbb{Q}}[G]=D^{2\times 2}\). Here the list is very short; it contains only three groups.