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On homologies in finite combinatorial geometries

Published online by Cambridge University Press:  17 April 2009

P.B. Kirkpatrick
P.B. Kirkpatrick
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
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Abstract

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Any subset π* of the set of all planes through a line in a finite projective space PG(m, q) determines a subgeometry G(π*) of the combinatorial geometry associated with PG(m, q). In this paper the geometries G(π*) of rank greater than three in which every line contains at least four points, are characterized in terms of the existence of a certain set of automorphism groups Γ(C, X); where X is a copoint and C a point not in X, and each non-trivial element of Γ(C, X) fixes X and every copoint through C and fixes C and every point in X, but no other point; and where Γ(C, X) acts transitively on the points distinct from C and not in X of some line through C. As a corollary of the main theorem we obtain a characterization of the finite projective spaces PG(m, q) with m ≥ 3 and q ≥ 3.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 1 , August 1975 , pp. 85 - 99
Copyright
Copyright © Australian Mathematical Society 1975

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