Subplanes of a translation plane

In this article we determine conditions for a finite translation plane π in order to contain subplanes and derivation sets. We study the effects of transposition on affine subplanes of π   introducing a geometric procedure to construct, starting from a Baer subplane π0π0 of π  , one π0′ lying in the transpose plane  πtπt of π  . We prove that, in general, π0′ is isomorphic neither to π0π0 nor to the transpose π0t of π0π0. Also, in the semifield case, we prove that some Knuth derivatives of the recently discovered Budaghyan–Helleseth semifield planes are derivable and hence derive to a strict translation plane. Our analysis further shows that any plane coordinatized by a commutative Albert semifield may be embedded in a Budaghyan–Helleseth plane.