A class of unitals of order q which can be embedded in two different planes of order \(q^ 2\) (Q1091606)

For every prime power \(q>2\) there exists a unital U of order q which is embeddable in two non-isomorphic projective planes of order \(q^ 2\), namely the Hall plane \(\Pi_ H(q^ 2)\) and its dual projective plane, respectively. The proof of this theorem is based upon the approach to construct translation planes of order \(q^ 2\) within PG(4,q) due to \textit{J. André} [Math. Z. 60, 156-186 (1954; Zbl 0056.385)] and a paper of \textit{F. Buekenhout} [Geom. Dedicaa 5, 189-194 (1976; Zbl 0336.50014)] on unitals in finite translation planes.