On spreads in PG(3,2s) that admit projective groups of order 2s

Let Γ be a spread in = PG(3, q); thus Γ consists of a set of q2 +1 mutually skew lines that partition the points of . Also let Λ be the group of projectivities of that leave Γ invariant: so Λ is the “linear translation complement” of Γ, modulo the kern homologies. Recently, inspired by a theorem of Bartalone [1], a number ofresults have been obtained, in an attempt to describe (Γ, Λ) when q2 divides |Λ|. A good example of such a result is the following theorem of Biliotti and Menichetti [3], which ultimately depends on Ganley's characterization of likeable functions of even characteristic [5].

Bireflectionality in Classical Groups

The motion groups of the real Euclidean plane and of the elliptic plane, the group of projectivities of a line, the projective general linear group PGL2(K), some orthogonal groups O3(K, Q) with char K = 2 (see [8]), are all bireflectional (zweispiegelig). There can be no doubt that bireflectional groups are of prime importance in any theory of groups that are generated by involutions. A brief look into F. Bachmann's book [1] gives convincing evidence.

The partitioning of an orthogonal group in six variables

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.

Conics and Orthogonal Projectivities In a Finite Plane

1. Introduction. The ternary orthogonal group of projectivities over a finite field leaves a non-singular conic ✗ invariant, but the geometry consequent thereupon does not appear to have been investigated. The group is isomorphic to a binary group of fractional substitutions over the same field and this fact may, since these binary groups and their subgroups are so well known, have inhibited projects to embark on a detailed description of the geometry of the ternary group.