DOUBLE CLASSES: A NEW CLASSIFICATION SCHEME FOR GROUP ELEMENTS

Publisher Summary Classification is one of the main goals of science, and the increasing importance of group theory makes a unified classification scheme for group elements desirable. This scheme contains the well-known cosets, double cosets, classes of conjugacy, and subclasses as special cases. The chapter presents an investigation of the new equivalence relations, which are expected in the framework of double classes. As double classes are orbits, their elements have conjugate stabilizers. If the latter are finite, they have a constant repetition frequency. The well-known classification schemes lead to two extreme cases. For double cosets, there are two completely uncorrelated symmetry groups in range and image spaces, and P, which is the group of operators, is a direct product of these two groups. For subclasses, there is a totally correlated symmetry; that is, the same transformation has to be carried out on range and image.