ABSTRACT
The concept of cycle index was first discovered by Howard Redfield in 1927. The cycle index formulas of various group actions have been computed since then by various authors. The formulas have been used to count graphs in mathematics and chemical compounds in chemistry. Therefore cycle index is a very useful tool in enumeration. It is widely applied in other fields of study like biology and jewelry industry. The cycle index of the symmetric group acting on ordered pairs, triples and on -ordered element subset has so far been done. The cycle index of dihedral and cyclic groups acting on both ordered and unordered pairs and triples has also been done, but cycle index of cyclic and dihedral groups acting on the Cartesian product of * + has not so far been done. In this project, we shall derive the cycle index formulas of the cyclic, ( ) and the dihedral, ( ) groups acting on the cartesian products of , where * +, for both n even and odd. In each case a number of examples have been worked out in details to illustrate the obtained results.
