Rank three affine planes

Kallaher, Michael J.

doi:10.1090/S0002-9939-1972-0313929-2

Rank three affine planes
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by Michael J. Kallaher

Proc. Amer. Math. Soc. 36 (1972), 79-86

DOI: https://doi.org/10.1090/S0002-9939-1972-0313929-2

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Abstract:

A permutation group has rank 3 if it is transitive and the stabilizer of a point has exactly three orbits. A rank 3 collineation group of an affine plane is one which is a rank 3 permutation group on the points. Several people (see [4], [7], [8], [12]) have characterized different kinds of affine planes using rank 3 collineation groups. In this article we prove the following: Let $\mathcal {A}$ be a finite affine plane of nonsquare order having a rank 3 collineation group which acts regularly on one of its orbits on the line at infinity. $\mathcal {A}$ must be either (i) a Desarguesian plane, (ii) a semifield plane, or (iii) a generalized André plane.

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