On Spreads Admitting Projective Linear Groups

Jha, Vikram ; Kallaher, Michael J.

Canadian journal of mathematics, Tome 33 (1981) no. 6, pp. 1487-1497

Résumé

In [8] Jha raised the following problem.(*) Let Γ be a spread whose components are subspaces of V 2n (GF(q)). Suppose G ≦ Aut Γ leaves a set of q + 1 components invariant while acting transitively on Γ\Δ.Find the possibilities for Γ or, more generally, the possibilities for (G, Γ, n, q).Many special cases of (*) have been settled. For instance, Cohen et al [1] have shown that if G fixes two non-zero points of V, that do not both lie in the same component of Γ, then Γ is the spread associated with either a Hall plane or the Lorimer-Rahilly plane of order 16 (LR-16) [14], [18].Another such result is given in [8]; there it is shown that if q is a prime number and G is a one-dimensional projective unimodular group then Γ is the spread associated with one of the following translation planes: (1) the Desarguesian planes of order 4, 8, or 9; (2) the nearfield plane of order 9; (3) LR-16; (4) the translation plane JW-16, obtained by transposing the slope maps of LR-16 [19].

Détail

DOI : 10.4153/CJM-1981-114-6

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