Über die Projektivitätengruppen von Semikörperebenen gerader Ordnung. (About groups of projectivities of semifield planes of even order)
By Th. Grundhöfer, the group of projectivities \(\Pi\) of any non- desarguesian (projective) translation plane P of order n contains the alternating group on \(n+1\) letters. Motivated by examples of Grundhöfer, Longwitz and Herzer, \textit{Th. Grundhöfer} [Result. Math. 6, 163-182 (1983; Zbl 0532.51002)] was led to conjecture that \(\Pi\) equals the symmetric group if the order of P is odd, and the alternating group if it is even. In the present note the author determines the group of projectivities \(\Pi\) of translation planes over some proper semifields (i.e. distributive quasifields) S of even order. He shows that, according to Grundhöfer's conjecture, \(\Pi\) consists only of even permutations, provided S is commutative or if S is of Knuth type II, III, or IV and \(| S| >16.\) On the other hand, proving the conjecture above wrong, he gives some examples of semifields of even order the group of projectivities of planes over which equals the symmetric group. In particular, the groups of projectivities of the two proper semifield-planes of order 16 contain odd permutations.
scientific article; zbMATH DE number 4111440
scientific article