Collineation groups preserving a unital in a projective plane of even order
Let \(\Pi\) be a finite projective plane of order \(m^ 2\); a unital \({\mathcal U}\) embedded in \(\Pi\) is a substructure of \(\Pi\) which forms a \(2-(m^ 3+1,m+1,1)\) design, and in particular a ``classical'' unital consists of absolute points and nonabsolute lines of a Hermitian polarity in a Desarguesian plane of order \(m^ 2.\) In this work the authors prove that, if \(\Pi\) is of even order \(m^ 2\) and admits a collineation group G which preserves a unital \({\mathcal U}\), such that G is transitive on the points of \({\mathcal U}\) and the socle of G has even order, then G contains involutory elations: in this case \(\Pi\) is Desarguesian, \({\mathcal U}\) is classical and if \(m>2\), \(PSU(3,m^ 2)\leq G\leq P\Gamma U(3,m^ 2).\) In this way as corollary the authors obtain that, if G is primitive on the points of \({\mathcal U}\), then \(\Pi\) is Desarguesian, \({\mathcal U}\) is classical and, if \(m>2\), \(PSU(3,m^ 2)\leq G\leq P\Gamma U(3,m^ 2)\).
scientific article; zbMATH DE number 4136872