Bilateral Classes

We introduce ’’bilateral classes’’, a new concept for classifying the elements of groups. Bilateral classes are orbits in a group G under the action of any given subgroup of the direct product G×G. The classification concept presented here encompasses conjugacy classes, cosets, double cosets, and Ree’s σ-classes as particular cases. It has an interpretation as a classification scheme for bijections between G-sets under the aspect of symmetry equivalence due to symmetries in both the domain and the range. While double cosets and conjugacy classes correspond to the case of no or complete correlation between operations of the two symmetry groups, our concept also covers the general case of partial correlation. The scope of generalization corresponds to applications in physics. Expressions for the number of bilateral classes are given.