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Collineation Groups Containing Perspectivities

Published online by Cambridge University Press:  20 November 2018

Peter Dembowski
Peter Dembowski*
Affiliation:
Universität Frankfurt am Main
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Let P be a projective plane of finite order n and Γ a group of collineations of P. Gleason (6) and Wagner (10) have shown that if every point of P is the centre, and every line the axis, of a non-trivial perspectivity in Γ, then Γ contains a subgroup of order n2 which consists entirely of elations. It then follows that either P or its dual is a translation plane with respect to at least one line; in fact if Γ has no fixed elements, then P is desarguesian and Γ contains all elations of P. It was shown by Piper (7) and Cofman (4) that the hypotheses of Gleason and Wagner can be relaxed in certain cases, while the same conclusions hold.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 924 - 937
Copyright
Copyright © Canadian Mathematical Society 1967

References

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