Correspondences and the theory of groups

Abstract

The object of this paper is to bring together several points connected with the general theory of correspondences and continuous groups, and to apply them to the theory of screws. Although the several results are in general not new, it seems of interest to give the accompanying presentation of the subject, as it furnishes an excellent examiple of the way in which the theory of continuous groups underlies the whole theory of correspondences. t The first section is devoted to general theory. Use is made of the theorem of LIE j that if we have a continuous group in n variables together with an invariant equation system inlvolving m paramneters, then a, group of the parameters' exists which is isomorphic with the given group, and it is pointed out that this theorem is fundamental iii all correspondences. ? The correspondence established is that between a P,M and a P n Contact transformationl is the particular case when mn n. The screw geometry is developed from the projective group in three dimensions together with the system of equations which define a general straight line. The general theory leads at once to two important results in conniection with the theory of groups: 1) The genercal continuous confornmal group in fourc dimensions is simply isomorphic with the generalprojective group in three. 2) Both these groups are simply isomorphic with the continuous projective group infive dinmensions which leaves a giv'en quadric invariant. There follows an imnmediate generalization of part of the second theorem. We have in fact the following: 3) The general conformal group in space of n dimensions is simply isomorphic with the projective group in space of n + 1 dimensions which preserves a given quadric. These three results are due to KLEIN. Some slight differences appear