ABSTRACT
The idea of isolating translation planes of order q2 admitting collineation groups in the translation complement of order q2 probably originated with Bartolone [100], who actually looked at such planes admitting larger groups of order q2(q − 1). Of course, any semifield plane of order q2 with kernel GF (q) admits such a group. But, there are also the interesting planes of Walker and Betten that also admit groups of these orders. The Lu¨neburg-Tits translation planes of order q2 admit Sz(q) as a collineation group in the translation complement, which acts doubly transitively on the infinite points and the stabilizer of an infinite point (∞) has the required order. Bartolone’s theorem states that under certain assumptions, these are exactly the possible translation planes admitting such groups.
