On semi-field planes of even order (Q790422)

The author extends a result of the reviewer and R. A. Liebler. Let \(\pi\) be a finite semifield plane of dimension d over its kernel (left nucleus \(K=GF(q)\), where \(q=p^ k\) with p a prime, and let A be the autotopism group of \(\pi\). The integer u is the number of orbits of A on the set \(\Omega\) of points of \(\pi\) not on the x- and y-axes. The reviewer has conjectured that \(u\geq 5\) if \(\pi\) is not desarguesian, and he has proven that \(u=1\) if and only if \(\pi\) is desarguesian [the reviewer, Arch. Math. 26, 436-440 (1975; Zbl 0314.50011)]. The reviewer and R. A. Liebler [Geom. Dedicata 8, 13-30 (1979; Zbl 0403.51010)] proved that \(u\geq 5\) if both p and d are odd integers. The present paper extends this result to the case where \(p=2\) and 4\(| d\). (The author assumes 4\(| kd\), but it can be shown that it is sufficient to assume 4\(| d)\).