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Generalised Elations

Generalised Elations KLAUS VEDDER a be a collineation of a finite projective plane n, and let F(<x) denote the Let structure consisting of all those points and lines of n which are fixed under a. F(a) contains the same number of points and lines [1]. Since F(a) is a closed configuration, it is one of the following four structures: (ii) F(<x) is a subplane of n, in which case a is called planar, (iii) F(a) contains an antiflag (C,/) and all the remaining lines of F(jx) contain C, while all the remaining points of F(a) are on / , (iv) the points of F(a) lie on a line/, the lines are concurrent in a point C on/ . The line/is called the axis of a, the point C its centre. Baer [1] named a collineation a with F(a) being of type (iii) a generalised homology, those of type (iv) generalised elations. So an elation (homology) is a generalised elation (homology) which fixes every point on its axis. If F(a) contains just two points, then a is both a generalised elation and a generalised homology. To exclude this possibility I will, henceforth, assume that a generalised elation fixes at least three points http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Generalised Elations

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/18.6.573
Publisher site
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Abstract

KLAUS VEDDER a be a collineation of a finite projective plane n, and let F(<x) denote the Let structure consisting of all those points and lines of n which are fixed under a. F(a) contains the same number of points and lines [1]. Since F(a) is a closed configuration, it is one of the following four structures: (ii) F(<x) is a subplane of n, in which case a is called planar, (iii) F(a) contains an antiflag (C,/) and all the remaining lines of F(jx) contain C, while all the remaining points of F(a) are on / , (iv) the points of F(a) lie on a line/, the lines are concurrent in a point C on/ . The line/is called the axis of a, the point C its centre. Baer [1] named a collineation a with F(a) being of type (iii) a generalised homology, those of type (iv) generalised elations. So an elation (homology) is a generalised elation (homology) which fixes every point on its axis. If F(a) contains just two points, then a is both a generalised elation and a generalised homology. To exclude this possibility I will, henceforth, assume that a generalised elation fixes at least three points

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1986

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