Hyperbolic ovals in finite planes
Let \(n = p^{2h}\) be a square power of an odd prime \(p\), and \(GF(n)\) the finite field of order \(n\). The triple \((N(n), +, \odot)\), where \(N(n)\) is the set of the elements of \(GF(n),\) and \(+, \odot\) are two operations on \(N(n)\) such that \(+\) is the same as the addition in \(GF(n)\), and the multiplication \(\odot\) is defined by \[ x \odot y = \begin{cases} xy & \text{for square } x\in GF(n), \\ x y^{p^h} & \text{for non-square } x\in GF(n), \end{cases} \] is a nearfield of dimension \(2\) over its center. Denote such a nearfield with \(N(n)\). Let \(\alpha\) be the affine plane of order \(n\) coordinatized by the above nearfield \(N(n)\). For any \(a, b, c, d\in GF(n)\) such that \(c\neq 0\) and \(a d - b c \neq 0\) let \[ \Omega_{a,b,c,d} = \biggl\{P(x, y)\in \alpha \, \left| \, x\in GF(n),\;x\neq -d/c,\;y = \frac{a x + b}{c x + d}\right.\biggr\}. \] Clearly \(\Omega_{a,b, c, d}\) has size \(n - 1\). \textit{L. A. Rosati} [Boll. Unione Mat. Ital. (4) 4, 463--467 (1971; Zbl 0229.50027)] proved the following result. Theorem. \(\Omega_{a,b,c,d}\) is a hyperbolic oval whose infinite points are \(X_\infty\) and \(Y_\infty\). Such hyperbolic ovals are equivalent under the collineation group of \(\pi\). In the paper under review the authors use the term ``Rosati oval'' to indicate \(\Omega_{a,b,c,d}\) and they give a characterization of such an oval. They prove the following result. Theorem. Let \(\Omega\) be a transitive hyperbolic oval in an affine translation plane \(\pi\) of odd order \(n\). Let \(D\) be a dihedral collineation group of order \(2(n - 1)\) of \(\pi\). If \(D\) preserves \(\Omega\) and acts transitively on the affine points of \(\Omega\), then one of the following cases occurs. i. \(\pi\) is the Desarguesian plane and \(\Omega\) is a conic. All non-central involutions are affine homologies, or ii. \(\pi\) is the regular nearfield plane of dimension \(2\) over its center and \(\Omega\) is a Rosati oval. The group \(D\) has \(\frac{1}{2}(n - 1)\) affine homologies and the same number of Baer involutions.
scientific article; zbMATH DE number 2091520