On the Structure of Finite Continuous Groups

Abstract

The structure of a finite continuous group is determined in part by the characteristic equation of the that is, the characteristic equation of the general infinitesimal transformation of the adjoint group of the given group. It is the purpose of this paper to show how certain properties of a group of linear homogeneous transformations can be obtained at once from the characteristic equation of the general infinitesimal transformation of the group, and thus how the type of structure is in part determined immediately from the infinitesimal transformations of such a group without the determination of the adjoint group. In ? 1 I show how to find the structural constants, and thus the roots of the characteristic equation, of the general infinitesimal transformation of the adjoint group of the general linear homogeneous group relative to a given subgroup; in ? 2 I find similar results for the special linear homogeneous group relative to a given subgroup. These results lead to certain relations, given in ? 3, between the characteristic equation of a group of linear homogeneous transformationcs and the characteristic equation of the general infinitesimal transformation of the given group. By the aid of these relations I establish theorems I-X by which certain properties of such a group can be at once determined from the general infinitesimal transformation of the group. Section 1.