Translation plane

In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of dualization and/or derivation.

In a projective plane, let $P$ represent a point, and $l$ represent a line. A central collineation with center $P$ and axis $l$ is a collineation fixing every point on $l$ and every line through $P$ . It is called an elation if $P$ is on $l$ , otherwise it is called a homology. The central collineations with center $P$ and axis $l$ form a group. A line $l$ in a projective plane $Π$ is a translation line if the group of all elations with axis $l$ acts transitively on the points of the affine plane obtained by removing $l$ from the plane $Π$ , $Π l$ (the affine derivative of $Π$ ). A projective plane with a translation line is called a translation plane.

The affine plane obtained by removing the translation line is called an affine translation plane. While it is often easier to work with projective planes, in this context several authors use the term translation plane to mean affine translation plane.

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