Null systems in projective space

If P is an (abstract) w-dimensional projective space, then we define a polarity in P as a correspondence p associating with every point Q in P a hyperplane Q and with every hyperplane h in P a point h in such a way that : (i) 0 = Q for every point () and A = A for every hyperplane h. (ii) The point Q is on the hyperplane h if, and only if, the hyperplane Q passes through the point h. I t is an immediate consequence of (i) that polarities are 1:1 correspondences. We shall term p a null-polarity if the polarity p has the additional property that : (iii) Every point Q is on the corresponding hyperplane Q, and consequently every hyperplane h passes through the corresponding point h. Extending a result of Veblen and Young, R. Brauer has shown that the existence of a null-polarity in P implies that the number n of dimensions of P is odd, and he has connected the null-polarities with the so-called null-systems, provided P is the w-dimensional projective space over a commutative field of coordinates. I t is the object of the present note to show that this last hypothesis may be omitted ; more precisely we are going to show that if the dimension of P is greater than 1, then the existence of a null polarity is equivalent to the fact that P is of odd dimension and is a projective space over a commutative field of coordinates. If P is a projective space of dimension 1, then the hyperplanes are points too. The identity transformation on the points of the line P is therefore the null-polarity of P . For this reason we shall assume throughout the remainder of this note that P be of dimension greater than 1. The case of a projective plane P has to be treated separately from the others, since the Theorem of Desargues need not hold true in a projective plane, though it is true for all the higher-dimensional projective spaces. A projective plane is a system of points and lines such that any two different lines meet in one and only one point, any two different