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On Spreads Admitting Projective Linear Groups

Published online by Cambridge University Press:  20 November 2018

Vikram Jha  and
Michael J. Kallaher
Vikram Jha
Affiliation:
Glasgow College of Technology, Glasgow, Scotland
Michael J. Kallaher
Affiliation:
Washington State University, Pullman, Washington
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In [8] Jha raised the following problem.

(*) Let Γ be a spread whose components are subspaces of V2n(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].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 6 , 01 December 1981 , pp. 1487 - 1497
Copyright
Copyright © Canadian Mathematical Society 1981

References

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