Translation Planes of Dimension Two with Odd Characteristic

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A translation plane of dimension d over its kernel K = GF(q) can be represented by a vector space of dimension 2d over K. The lines through the zero vector form a “spread”; i.e., a class of mutually independent vector spaces of dimension d which cover the vector space.The case where d = 2 has aroused the most interest. The more exotic translation planes tend to be of dimension two; a spread in this case can be interpreted as a class of mutually skew lines in projective three-space.The stabilizer of the zero vector in the group of collineations is a group of semi-linear transformations and is called the translation complement. The subgroup consisting of linear transformations is the linear translation complement.