The invariants of a finite collineation group in five dimensions

Abstract

In 1914 Mitchell (13) underook the determination of the finite primitive collineation groups in more than three dimensions which are generated by homologies. Among these groups are five which are remarkable as not being members of an infinitely extended series, and these five isolated groups fall naturally into two sets. One set comprises three groups of orders 27.34:5, 29.34.5.7 and 213.35.52.7, belonging respectively to space of five, six and seven dimensions, the centres of the generating homologies of the smaller groups being among those of the larger ones; these groups had been considered earlier by Burnside (3) as groups of linear substitutions with rational coefficients, and both the groups and the configurations formed by the centres of the generating homologies are closely related to polytopes in Euclidean space of six, seven and eight dimensions first described by Gosset (9) and later studied in detail by Coxeter (6, 7). The two remaining groups, of orders 26.34.5 and 28.36.5.7, belong respectively to four and five dimensions, and the centres of the forty-five generating homologies of the former appear among the centres of the 126 generating homologies of the latter. The smaller group is the well-known simple group of order 25920, and is probably the most extensively studied of all special groups of finite order. Its representation as a collineation group in four dimensions was first obtained by Burkhardt (2), who determined the invariants of the group; the simplest of these is a quartic primal with nodes at the 45 centres of homologies, which was later studied by Coble (5) and myself (14). The geometry of the configuration formed by these 45 nodes has been investigated in detail by Baker (1), whose results I have used elsewhere (15) to classify the operations of the group and obtain their distribution into conjugate sets.