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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
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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

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