Semifield planes of order \(p^ 4\) that admit a \(p\)-primitive Baer collineation (Q1191039)

Let \(\pi\) be a semifield plane of order \(q^ 2\) and kernel containing \(GF(q)\). Then \(\sigma\) is a \(p\)-primitive Baer collineation if \(\sigma\) fixes a Baer subplane pointwise and \(| \sigma|\) is a \(p\)- primitive divisor of \(q-1\). A semifield plane of order \(p^ 4\) and kernel \(GF(p^ 2)\), where \(p\) is an odd prime, is called a \(p\)-primitive semifield plane if it admits a \(p\)-primitive Baer collineation. The author deals with some properties of \(p\)-primitive semifield planes and determines necessary and sufficient conditions for isomorphism within this class. The main result is: For any odd prime \(p\), there are \(({p+1\over 2})^ 2\) nonisomorphic \(p\)-primitive semifield planes of order \(p^ 4\). It is shown that of these, \({p+1\over 2}\) are Hughes-Kleinfeld semifield planes and one is a Dickson semifield plane. Also, the Boerner-Lantz semifield planes of order \(p^ 4\) are shown to be \(p\)-primitive semifield planes. Each of the remaining planes is either generalized twisted field plane or is a new plane. Particularly, this paper follows up \textit{Y. Hiramine, M. Matsumoto} and \textit{T. Oyama} [ibid. 24, No. 1, 123-137 (1987; Zbl 0646.51006)] and \textit{N. L. Johnson} [ibid. 25, No. 3, 519-530 (1988; Zbl 0707.51002)].