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Collineation Groups of Generalized André Planes

Published online by Cambridge University Press:  20 November 2018

David A. Foulser
David A. Foulser*
Affiliation:
University of Illinois at Chicago Circle, Chicago, Illinois
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In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.

The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 358 - 369
Copyright
Copyright © Canadian Mathematical Society 1969

References

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