Semifield planes of characteristic p admitting p-primitive Baer collineations (Q919265)

Let \(\Pi\) be a semifield plane of order \(q^ 2\) and kernel GF(q), where q is a prime power \(p^ r\). A p-primitive Baer collineation of \(\Pi\) is a collineation \(\sigma\) which fixes a Baer subplane pointwise and whose order is a p-primitive divisor of q-1. For the sake of this review call \(\Pi\) a special semifield plane if it has a p-primitive Baer collineation. The author in this article proves the following facts about special semifield planes: (i) For such planes the power q is a square (i.e., r is even); (ii) Given a semifield plane of order w and Ostrom dimension \(d\leq 2\) over its kernel a special semifield plane of order \(w^ 2\) can be constructed using the method of extension discovered by \textit{Y. Hiramine}, \textit{M. Matsumoto} and \textit{T. Oyama} [Osaka J. Math. 24, 123- 137 (1987; Zbl 0646.51006)]; (iii) Given a special semifield plane \(\Pi\) of order q the method of contraction discovered by Hiramine, Matsumoto, and Oyama can be applied to give a semifield plane of order \(\sqrt{q}\) and Ostrom dimension at most 2. Thus, special planes are very common in the class of semifield planes having Ostrom dimension 2 over their kernel.