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On Translation Planes which Admit Solvable Autotopism Groups Having a Large Slope Orbit

Published online by Cambridge University Press:  20 November 2018

Vikram Jha
Vikram Jha*
Affiliation:
University of Iowa, Iowa City, Iowa
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Our main object is to prove the following result.

THEOREM C. Let A be an affine translation plane of order qr ≧ q2 suchthatl∞, the line at infinity, coincides with the translation axis of A. Suppose G is a solvable autotopism group of A that leaves invariant a set Δ of q + 1 slopes and acts transitively on l∞ \ Δ.

Then the order of A is q2.

An autotopism group of any affine plane A is a collineation group G that fixes at least two of the affine lines of A; if in fact the fixed elements of G form a subplane of A we call G a planar group. When A in the theorem is a Hall plane [4, p. 187], or a generalized Hall plane ([13]), G can be chosen to be a planar group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 5 , 01 October 1984 , pp. 769 - 782
Copyright
Copyright © Canadian Mathematical Society 1984

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