Ovals and unitals in commutative twisted field planes
Let \(GF(q)\) be a Galois field of order \(q = p^{2n}\), with \(p\) an odd prime and \(n\) a positive integer, containing a subfield \(GF(d)\) such that \((-1)\) is not a \(d-1\) power in \(GF(q)\). The commutative twisted field \(Q(+,*)\) of order \(q\) may be obtained from \(GF(q)\) by replacing the multiplication in \(GF(q)\) with a new one defined by the rule : \((a^d+a)*(b^d+b) = a^db+ab^d\) (see section 5.3.7 of \textit{P. Dembowski}'s book ``Finite Geometries'' (1968; Zbl 0159.50001). Correspondingly, the affine plane \(\Pi\) coordinatized by \(Q(+,*)\) may be obtained from the desarguesian plane \(AG(2,q)\) over \(GF(q)\) by replacing lines not through \((\infty)\) with graphs of functions over \(GF(q)\). These functions are \(y = m^dx+mx^d+w\) where \(m, w \in GF(q)\). Let \(\Theta : x \leftarrow x^{\sqrt{q}}\) be the involutory automorphism of \(GF(q)\). According to [\textit{M. I. Ganley}, Geom. Dedicata 1, 103-116 (1972; Zbl 0241.50024)], Theorem 8, the mapping which turns the point \(P(u,v)\) into the line with equation \(y = u^\Theta*x-v^\Theta\) is then a unitary polarity whose absolute points form the unital \({\mathcal U} = \{(u,v) : v+v^\Theta-u^\Theta*u = 0\); \(u, v \in Q(+,*)\} \cup \{(\infty)\}\). The authors prove the following result determining the collineation group of \({\mathcal U}.\) Let \(\Phi\) be the collineation group of \(\Pi\) fixing the unital \({\mathcal U}\). Then (i) \(\Phi\) has order \(mq\sqrt{q}(\sqrt{q}-1)(\sqrt{d}+1)\), where \(q=p^m\), \(p\) a prime; (ii) \(\Phi\) has a normal subgroup \(M\) of order \(q\sqrt{q}\) that acts on the affine points of \({\mathcal U}\) as a sharply transitive permutation group; (iii) the stabilizer of an affine point of \({\mathcal U}\) under \(\Phi\) has a normal cyclic subgroup of order \((\sqrt{q}-1)(\sqrt{d}+1)\) with cyclic complement of order \(m\).
polarities in commutative twisted field planes
collineation groups of ovals and unitals
scientific article; zbMATH DE number 1369765