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Conjecture of D. R. Hughes Extended to Generalized André Planes

Published online by Cambridge University Press:  20 November 2018

M. L. Narayana Ra
M. L. Narayana Ra*
Affiliation:
University of Missouri, Columbia, Missouri
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In 1967 Foulser [1] defined a class of translation planes, called generalized André planes or λ-planes and discussed the associated autotopism collineation groups. While discussing these collineation groups he raised the following question:

“Are there collineations of a λ plane which move the axes but do not interchange them?”.

In this context, Foulser mentioned a conjecture of D. R. Hughes that among the André planes, only the Hall planes have collineations moving the axes without interchanging them. Wilke [4] answered Foulser's question partially by showing that the conjecture of Hughes is indeed correct. Recently, Foulser [2] has shown that possibly with a certain exception the Hall planes are the only generalized André planes which have collineations moving the axes without interchanging them. Our aim in this paper is to give an alternate proof, which is completely general, and is in the style of the original problem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 701 - 704
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Foulser, D. A., A generalization of Andre's systems, Math. Z. 100 (1967), 380–395.Google Scholar
2. Foulser, D. A., Collineation groups of generalized Andre planes, Can. J. Math. 21 (1969), 358–369.Google Scholar
3. Marshall, Hall Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
4. Wilke, F. W., A class of translation planes and a conjecture of D. R. Hughes, Trans. Amer. Math. Soc. 45 (1969), 223–232.Google Scholar