Polya' Theory - Book Report/Review Example

Group action Consider a necklace with four colored beads with the possibility of each bead being green or blue. The necklace can be flipped over and the beads are able to rotate around the necklace. The necklace may have 16 different arrangements. However, the rotation of the beads may yield similar arrangements. Flipping the beads also results in similar necklace arrangements for different flip modes. These operations are called necklace rigid motions. Each necklace rigid motion permutes the elements of the necklace to affect the arrangement (Po?

lya and Tarjan et al., 2010). The rigid motions are represented by permutations of S or X. the S permutations are commonly used as they represent the four vital elements, which is advantageous over the X elements that are 16 in number. Suppose X represents a finite set like a set of beads and G a group of permutations of the finite set X or a symmetry group that is finite and acts on X. Therefore, G represents a particular group of permutations of the set of beads. If the beads make up an necklace with n number of beads, the rotational symmetry is relevant and G is the cyclic group, C.

supposing that the set of colors of the finite beads is denoted Y, YX denotes the set of colored arrangements of the beads (Po?lya and Tarjan et al., 2010). This means Y is a function of X with the number of colors found by |Y| =t. representing elements of G as permutations of X requires a notation that is much more extensive. A 900 anticlockwise rotation of G, the permutation (1)(2,3,4,5)(6,7,8,9)(10,11)(12,13,14,15)(16) on X could be denoted by writing ?2. In comparison, the use of the four-value system will illustrate the rigid motion as (1234).

According to the theorem, the number of orbits of G of colored arrangements is calculated using: . Suppose S is a collection of objects and R a set of elements like bead colors. Describing R as colors of S means the assignment of unique colors to each of the elements of S. this relation is interpreted as ; f : S > R. if the value of |S| is n and |R| is m the total distinct colorings will be mn . for a set Y and group G, an action of G on Y creates the relationship G ? Y > Y. this satisfies two properties of the theory illustrated below. 1. (g1g2)(y) = g1(g2(y)) for all g1, g2 € G and y € Y 2. e(y) = y for all y € Y, where e represents the identity element in G Burnside’s theorem This theorem is applied in the counting of the patterns when a group is acting on a set of colorings.

The focus lies in the count of the number of orbits when the group is acting on the set. Supposing a group G is acting on a set of elements Y. for each element ? € G in the set, it is fixed by ? as Fix(?). Generally this implies Fix(?) = {y € Y | ?(y) = y}. Applying the specifics of a necklace similar to the one in the previous theorem, the rigid motions of a square, G, that are acting on the set X, a sequence is developed (Po?lya and Tarjan et al., 2010). The enumeration relies on the on the notation ?

i and is listed as Fix(?i)for each value of ?i| € G. in the case of group G and set Y, for each element of y € Y, the subgroup of the elements of G that fix y are denoted as Stab(y) which implies Stab(y) ={? € G| ?(y) = y}. The sum of all the values of |Stab(y) | and |fix(?i) | for the necklace is 48. This illustrates that the sum of these values for other objects that are subjected to the theory is the same. The equality is guaranteed in the lemma below. ?|Fix(?)| =?