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A transitive self-polar double-twenty of planes

Part of: Geometry Linear incidence geometry

Published online by Cambridge University Press:  09 April 2009

P. B. Kirkpatrick
P. B. Kirkpatrick
Affiliation:
School of Mathematics and Statistics University of Sydney, NSW 2006, Australia
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Abstract

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We demonstrate the existence, in the 5-dimensional projective space over any field J in which 1 + 1 ≠ 0 and −1 is a square, of a non-degenerate double-twenty of planes (ℋ, K) with the property that there is a group of collineations which acts transitively on ℋ ∪ K while each element of the group either maps ℋ onto itself and K onto itself or else swaps ℋ with K. If there is an involutory automorphism of J which swaps the two square roots of −1, then (ℋ, K) is also self-polar (with respect to a unitary polarity). We describe some of the geometry (in both 5-dimensional and 3-dimensional space) associated with the double-twenty (ℋ, K) and its group.

MSC classification

Secondary: 51A45: Incidence structures imbeddable into projective geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 2 , October 1996 , pp. 249 - 257
Copyright
Copyright © Australian Mathematical Society 1996

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