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On Baer Involutions of Finite Projective Planes

Published online by Cambridge University Press:  20 November 2018

Judita Cofman
Judita Cofman*
Affiliation:
Imperial College of Science and Technology, London, England
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1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.

While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 878 - 880
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. J., André, Uber Perspektivitàten in endlichen projektiven Ebenen, Arch. Math. 6 (1954), 29–32.Google Scholar
2. Baer, R., Projectivities of finite projective planes, Amer. J. Math. 69 (1947), 653–684.Google Scholar
3. Hall, M. Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
4. Piper, F. C., Collineation groups containing elations. I, Math. Z. 89 (1965), 181–191.Google Scholar
5. Wagner, A., On perspectivities of finite projective planes, Math. Z. 71 (1959), 113–123.Google Scholar