Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T16:27:34.385Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Projective Planes that Admit a Strongly Irreducible Collineation Group

Published online by Cambridge University Press:  20 November 2018

Chat Yin Ho
Chat Yin Ho*
Affiliation:
University of Toronto, Toronto, Ontario
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.

This paper studies how coding theory and group theory can be used to produce information about a finite projective plane π and a collineation group G of π.

A new proof for Hering's bound on |G| is given in 2.5. Using the idea of coding theory developed in [9], a relation between two rows of the incidence matrix of π with respect to a tactical decomposition is obtained in 2.1. This result yields, among other things, some techniques in calculating |G|, and generalizes a result of Roth [16], [see 2.4 and 2.5].

Hering [7] introduced the notion of strong irreducibility of G, that is, G does not leave invariant any point, line, triangle or proper subplane. He showed that if in addition G contains a non-trivial perspectivity, then there is a unique minimal normal subgroup of G. This subgroup is either non-abelian simple or isomorphic to the elementary abelian group Z3 × Z3 of order 9.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985 , pp. 579 - 611
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Bloom, D. M., The subgroups of PSL(3, q) for odd q, Trans AMS 127 (1967), 150–178.Google Scholar
2. Dembowski, P., Finite geometries (Springer, 1972).Google Scholar
3. Feit, W., The current situation in the theory of finite simple groups, Acte congre intern. math. 1 (1970) 55–93.Google Scholar
4. Goncalves, A. and Barboza, D. F., The projective plane of order 47 and 53, to appear.Google Scholar
5. Goncalves, A. and Pomareda, R., Pianos projectivos de orden 17, Notas de la Soc. Mat. de Chile 1 (1981).Google Scholar
6. Gorenstein, D., Finite groups (Harper and Row, 1968).Google Scholar
7. Hering, C., On the structure of finite collineation groups of projective planes, Abh. Math. Sem. Univ. Hamburg 49 (1975), 155–182.Google Scholar
8. Hering, C., Finite collineation groups of projective planes containing non-trivial perspectivities, to appear.Google Scholar
9. Hering, C., On codes and projective design, Kyoto Univ. Math. Research Inst. Sem. 344 (1979), 26–60.Google Scholar
10. Hering, C. and Walker, M., Perspectivities in irreducible collineation groups of projective planes I, Math. Z. 155 (1977), 95–101; II, J. of Statistic Planning Inference 3 (1979), 151–177.Google Scholar
11. Hughes, D. and Piper, F., Projective planes (Springer, 1973).Google Scholar
12. Huppert, B., Endliche Gruppen I (Springer, 1967).CrossRefGoogle Scholar
13. Kantor, W., On unitary polarities of finite projective planes, Can. J. Math. 23 (1971), 1060–1077.Google Scholar
14. Kantor, W., On the structure of collineation groups of finite projective planes, Proc. London Math. Soc. 32 (1976), 385–402.Google Scholar
15. Pickert, G., Projektive Ebenen (Springer, 1975).CrossRefGoogle Scholar
16. Roth, R., Collineation groups of finite projective planes, Math. Z. 83 (1964), 409–421.Google Scholar
17. Filho, A. M. Serra, Pianos projectivos de ordem prima, Doctoral Thesis, Univ. of Brasilia (1981).Google Scholar