The Existence of Line Involutions of Order Greater than Three Possessing a Linear Complex of Invariant Lines

Introduction. In a recent paper [l] attention was called to a new family of line involutions in 53 furnishing examples of involutions of all orders, m, ^4 with complexes of invariant lines of all possible orders, i, from 2 up to the maximum, [(»i + l)/2]. Since involutions of all orders without a complex of invariant lines are known to exist, and since examples of all possible involutions of order <4 are known, the only involutions for which existence examples remain to be supplied are those whose orders are 2:4 and whose invariant lines form a linear complex. It is the purpose of this note to define a class of involutions having these properties, thus establishing the existence of line involutions corresponding to every admissible set of characteristics im, », i, k) [l]. In our development we shall work exclusively on the nonsingular Vl in S6 into whose points the lines of S3 are mapped in a 1:1 way by the well known interpretation of the Plücker coordinates of a line in S3 as point coordinates in S6.