Irreducible collineation groups fixing an oval (Q1969658)

A long standing problem in the theory of projective planes is the determination of collineation groups fixing an oval. Of course, to make headway additional properties on the group must be assumed. In this article, the authors assume the group is irreducible on the plane; that is, the group fixes no point, line, or triangle. Irreducible collineation groups of projective planes have been extensively studied. This study was initiated by \textit{C. Hering} [Abh. Math. Semin. Univ. Hamb. 49, 155-182 (1979; Zbl 0431.51002)] who gave a deep clarification theorem, and it has been extended by others. Hering's work has been very useful when the irreducible collineation group fixes interesting substructures such as ovals and unitals. This article studies irreducible collineation groups \(G\) fixing an oval in a projective plane of odd order and having \(|G|\) divisible by 4. The authors prove \(G\) must contain involutory homologies. It then follows from \textit{M. Biliotti} and \textit{G. Korchmaros} [Geom. Dedicata 57, No. 1, 73-89 (1995; Zbl 0836.51007)] that \(G\cong \text{PSL}(2,q)\) with \(g\geq 5\) or \(G\) contains a subgroup that is a central extension of \({\mathcal A}_6\) by a cyclic group of order 3. Furthermore, the group \(G\) is minimal (i.e., no subgroup of \(G\) is irreducible) if and only if \(G\cong \text{PSL}(2,q)\) with \(q\geq 5\) and \(q^2\not\equiv 1\pmod 5\).