Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T21:33:22.046Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Cohen, S. D., Ganley, M. J. and Jha, V., On transitive automorphism groups of finite quasifields, to appear in Archiv Der Math.Google Scholar
2. Hartley, R. W., Determination of the ternary collineation groups whose coefficients lie in GF(2n), Ann. Math. 27 (1925-26), 140–158.Google Scholar
3. Harris, M. E. and Hering, C., On the smallest degrees of projective representations of the group PSL(n, q), Can. J. Math. 23 (1971), 90–102.Google Scholar
4. Hering, C., On shears of translation planes, Abh. Math. Sem. Hamb. 27 (1972), 258–268.Google Scholar
5. Hughes, D. R. and Piper, F. C., Projective planes, (Springer-Verlag, Berlin, 1973).Google Scholar
6. Huppert, B., Endliche Gruppen I, (Springer-Verlag, Berlin, 1967).Google Scholar
7. Jha, V., On tangentially transitive translation planes and related systems, Geom. Ded. 4 (1975), 457–483.Google Scholar
8. Jha, V., A characterization of some translation planes which admit unimodular linear fractional groups, to appear.Google Scholar
9. Jha, V., On subgroups and factor groups of GL(n, q) acting on spreads with the same characteristic, to appear.Google Scholar
10. Jha, V., On finite semitransitive translation planes, to appear.Google Scholar
11. Johnson, N. L. and Ostrom, T. G., Translation planes of characteristic 2 in which all involutions are Baer, J. Alg. 54 (1978), 291–315.Google Scholar
12. Johnson, N. L. and Ostrom, T. G., The translation planes of order 16 that admit PSL(2, 7), J. Comb. Theory A26 (1979), 127–134.Google Scholar
13. Kallaher, M. J., On the Lorimer-Rahilly and the Johnson-Walker translation planes, Proc. of Conference in Finite Geometries, Sussex (1980), to appear.Google Scholar
14. Lorimer, P., A projective plane of order 16, J. Comb. Theory A16 (1974), 334–347.Google Scholar
15. Mitchell, H. H., Determination of ordinary and modular ternary linear groups, Trans. Amer. Math. Soc. 12 (1911), 207–242.Google Scholar
16. Ostrom, T. G., Finite translation planes, Lecture Notes in Mathematics 158 (Springer-Verlag, Berlin, 1970).Google Scholar
17. Ostrom, T. G., Elementary abelian 2-groups in finite translation planes, Arch. Math., to appear.Google Scholar
18. Rahilly, A., Some translation planes with elations which are not translations, Combinatorial Mathematics III, Lecture Notes in Mathematics 452 (Springer-Verlag, Berlin, 1970).Google Scholar
19. Walker, M., A note on tangentially transitive affine planes, Bull. Lond. Math. Soc. 8 (1976), 273–277.Google Scholar
20. Zsigmondy, K., Zur Théorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), 265–284.Google Scholar