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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 , , , ) have characterized different kinds of affine planes using rank 3 collineation groups. In this article we prove the following: Let si be a finite affine plane of nonsquare order having a rank 3 collineation group which acts regularly on one of itsdoi:10.2307/2039040 fatcat:z7jy3p53jzf67d5xe2fzkeks74