Abstract:
This study is carried out in order to investigate permutation groups, Graph theory and Polya's theory of counting with applications together with the relations between them. The key to this relationship is the celebrated Burnside lemma. Aimed to explain the aspects of group theory which are related to them. Moreover numearous groups of permutations and the cyclic structures of their elements together with the orbits of those elements are then used methods and scientific means to enumerate all the possible ways of colourings of a set. This is then used to prove polya’s enumeration theorem (PET). The most important results of this study to obtain some applications of permutation groups, Graph theory and Polya's theory of counting .
