On translation planes of order \(q^ 3\) that admit a collineation group of order \(q^ 3\)
The authors study translation planes of order \(k\), that admit a collineation group \(G\) of order \(k\) in the linear translation complement, for \(k = q^3\) (\(q\) -- a prime power). With the following additional assumptions all such planes are determined. Authors require that the kernel of a plane contains \(GF(q)\), a group \(G\) fixes one point at infinity and acts transitively on the remaining points at infinity, the elation group \(E\) of \(G\) is of order \(q\), and \(G/E\), is abelian. Authors show a general construction of spread sets which yields exactly all the planes in question. The definitions of spread make use of certain parameters of \(\text{GF} (q)\), so the construction is not really effective; to obtain a desired plane one must find concrete parameters which really yield a spread set. On the other hand, a matrix representation of the corresponding groups \(G\) and \(E\) is given. Next, the authors consider the special case of prime \(q\). Then \(E\) is of order \(q^3\) or the requirements on \(E\) follow from the remaining assumptions about \(G\), and a plane in question is either a semifield plane (of order \(q^3\)), or the Sherk plane of order 27.
scientific article; zbMATH DE number 806218