On The Cycle Indices of Cyclic and Dihedral Groups Acting on X^2

International Journal of Research and Scientific Innovation (IJRSI) | Volume VI, Issue XII, December 2019 | ISSN 2321–2705

On The Cycle Indices of Cyclic and Dihedral Groups Acting on X^2

Felix Komu^1*, Ireri N. Kamuti²

¹Department of Mathematics and Physics, Moi University, Kenya
²Department of Mathematics and Actuarial Science, Kenyatta University, Kenya
*Corresponding author

Abstract: An effective method of deriving the cycle indices of cyclic and dihedral groups acting on X^2, where X={1,2,…,n} is provided. This paper extents some results of Harary and Palmer(1973); Krishnamurthy (1985) and thoka et.al.(2015).

I. INTRODUCTION

The concept of cycle index was first done by Howard Redfield in 1927, however, his paper was overlooked but came to attention of Mathematicians long after his death in 1944. The concept was later rediscovered by Pölya in an independent study and applied his results to solve interesting combinatorial problems in chemistr as outlined by Donald Woods(1979).
The cycle index of Cyclic and Dihedral groups acting on X can be found in various books (Krishnamurthy, 1985; Harary and Palmer, 1973) respectively.
The cycle index of cyclic and dihedral groups acting on ordered pairs was computed by Muthoka et. al. in 2015.

II. DEFINITIONS AND PRELIMINARY RESULTS

This section outlines some definitions and established results that will be used throughout this paper.
Definition 1. A dihedral group is the group of symmetries of a regular n-gon. It has degree n and order 2n.
Definition 2. A cyclic group is a group of order nthat can be generated by a single element.

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