The nuclei and other properties of \(p\)-primitive semifield planes (Q1187602)

Let \(\pi\) be a semifield plane of order \(p^ 4\), where \(p\) is an odd prime, with kernel \(K\cong GF(p^ 2)\). Assume that \(\pi\) admits a Baer collineation whose order divides \(p^ 2-1\) but not \(p-1\). In this case we call \(\pi\) a \(p\)-primitive semifield plane. If \(S\) is the semifield coordinatizing \(\pi\), define as usual the right nucleus of \(S\) to be \(N_ r=\{d\in S:(xy)d=x(yd)\) for all \(x,y\in S\}\). Similarly define the left nucleus \(N_ l\) and the middle nucleus \(N_ m\). If by convention we assume quasifields are right quasifields, then the left nucleus of \(S\) is just the kernel and hence \(N_ l\cong GF(p^ 2)\). The author shows there are just two possibilities for the various nuclei of \(S\) under the above hypotheses. Namely, either \(N_ m=N_ r=N_ l\cong GF(p^ 2)\) or \(N_ m=N_ r\cong GF(p)\). It is also shown that if \(\pi\) is a \(p\)-primitive semifield plane, then the transposed plane \(\pi^ t\) (obtained by taking the transpose of each matrix in the spread set corresponding to \(\pi)\) is isomorphic to \(\pi\). Moreover, if the dual plane \(\pi^ D\) is also \(p\)-primitive, then \(\pi\) is a Hughes-Kleinfeld semifield plane.