Quadratic extensions of flag-transitive planes (Q1970074)

A finite affine plane \(\pi\) of order \(q^2\) which has a subplane \(\pi_0\) of order \(q\) is called a quadratic extension of a flag-transitive plane if it admits a collineation group \(G\) which leaves \(\pi_0\) invariant, acts transitively on the flags of \(\pi_0\), and acts transitively on the lines of \(\pi\) intersecting \(\pi_0\) in precisely one affine point. The authors prove that the only finite translation planes which are quadratic extensions of flag-transitive planes are the Desarguesian planes, the Hall planes, and the derived likeable Walker plane of order 25.