A necessary condition that two finite quasi-fields coordinatize isomorphic translation planes

Abstract

Given two quasi-fields it is usually easier to determine whether or not they are isotopic or anti-isotopic than it is to determine whether or not they can coordinatize isomorphic planes. The theorem given here has proved to be quite useful for this purpose. (See, for example, [2].) A translation plane wr may always be coordinatized, in the Hall sense [1], by a right quasi-field (Veblen-Weddenburn system). It is known that two quasi-fields which are either isotopic or anti-isotopic coordinatize isomorphic translation planes. This note gives a necessary condition that two finite quasi-fields which are neither isotopic nor anti-isotopic can coordinatize isomorphic translation planes. For the remainder of this note let F1, F2 be finite quasi-fields and 7rl, 72 the associated translation planes in the sense of Hall [1]. For i=1, 2 let Fi,p= {aFiaxo and (xy)a=x(ya)Vx, yEFi} =right nucleus of Fi and let Fi, = { a E Fi I a 0 and (xa)y = x (ay)} =middle nucleus of Fi. It is easy to see that for each aCFi, the mapping O3a: (x, y)-*(x, ya), (m)-*(ma), Yi=(oo)-*Yi is a perspectivity of 7r with center Yi and axis the line y =0. The correspondence a 1a iS an isomorphism between Fip and the set of Y -OiXi perspectivities of ri. Also, for each aEFiu, the mapping ya: (X, y)-*(xa, y), (m) -*(mLm-1), Yi-*Yi is a Xi-OiYi perspectivity of ri and the correspondence a ya is an isomorphism between Fi, and the set of XiO Yi perspectivities of ri. We will use the conventional notation Y= (oo ), 0 = (0, 0), X = (0). We will also use the symbol j A j to denote the cardinality of a set A.