ABSTRACT
If a group G acts on a set X, then X breaks up into the disjoint union of the various orbits of G. When X and G are finite, we can obtain formulas relating the number of elements in the orbits and stabilizers and in X, and for the number of fixed points of the elements of G. These formulas are useful in studying the structure of abstract finite groups and of symmetry groups. They also have applications to combinatorial problems. For any finite set Y, we shall denote the number of elements in Y by |Y|.
