Cycle Index - Example

Example

Consider the group G of rotational symmetries of a square in the Euclidean plane. Such symmetries are completely determined by the images of just the corners of the square. By labeling these corners 1, 2, 3 and 4 (consecutively going clockwise) we can represent the elements of G as permutations of the set X = {1,2,3,4}. The permutation representation of G consists of the four permutations (1 4 3 2), (1 3)(2 4), (1 2 3 4) and e = (1)(2)(3)(4) which represent the counter-clockwise rotations by 90°, 180°, 270° and 360° respectively. Notice that the identity permutation e is the only permutation with fixed points in this representation of G. As an abstract group, G is known as the cyclic group C₄, and this permutation representation of it is its regular representation. The cycle index monomials are a₄, a₂2, a₄, and a₁4 respectively. Thus, the cycle index of this permutation group is:

The group C₄ also acts on the unordered pairs of elements of X in a natural way. Any permutation g would send {x,y} → {xg, yg} (where xg is the image of the element x under the permutation g). The set X is now {A, B, C, D, E, F} where A = {1,2}, B = {2,3}, C = {3,4}, D = {1.4}, E = {1,3} and F = {2,4}. These elements can be thought of as the sides and diagonals of the square or, in a completely different setting, as the edges of the complete graph K₄. Acting on this new set, the four group elements are now represented by (A D C B)(E F), (A C)(B D)(E)(F), (A B C D)(E F) and e = (A)(B)(C)(D)(E)(F) and the cycle index of this action is:

The group C₄ can also act on the ordered pairs of elements of X in the same natural way. Any permutation g would send (x,y) → (xg, yg) (in this case we would also have ordered pairs of the form (x, x)). The elements of X could be thought of as the arcs of the complete digraph D₄ (with loops at each vertex). The cycle index in this case would be: