The partitioning of an orthogonal group in six variables

Abstract

An orthogonal group of projectivities in six variables is partitioned into 31 disjoint sets; 17 of these constitute a subgroup, of index 2 and order 3265920, isomorphic to a group already known in great detail in another representation. The ambient space is finite, of five dimensions; its points fall into three batches of 126, 126, 112, these last composing the quadric invariant under the group. Each projectivity permutes the points of each batch. The cyclic decompositions of all these permutations are listed in two tables, as also are the numbers of projectivities in the various sets of the par­titioning, together with any spaces composed of points that are invariant. The prime purpose is to demonstrate with what rapidity and uniformity much of the work is done once the results for an analogous group with one variable fewer are on record; indeed 14 of the 17 sets mentioned above fall out in this way. There are numerous allusions to two earlier papers wherein essential information is assembled.