Properties of Suborbits of the Dihedral Group D<sub>n</sub> Acting on Ordered Subsets

Abstract

A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, emphasizing the permutation aspect, so that the algebraic structure has a wider application among other algebras. Such structures not only reveal connections between different areas in Mathematics but also make use of results in one area to suggest conjectures and also prove results in a related area. The structure (G, X) is a transitive permutation group G acting on the set X. Investigations on the properties associated with various groups acting on various sets have formed a subject of recent study. A lot of investigations have been done on the action of the symmetric group Sn on various sets, with regard to rank, suborbits and subdegrees. However, the action of the dihedral group has not been thoroughly worked on. This study aims at investigating the properties of suborbits of the dihedral group Dn acting on ordered subsets of X={1,2,...,N}. The action of Dn on X[r], the set of all ordered r-element subsets of X, has been shown to be transitive if and only if n = 3. The number of self-paired suborbits of Dn acting on X[r] has been determined, amongst other properties. Some of the results have been used to determine graphical properties of associated suborbital graphs, which also reflect some group theoretic properties. It has also been proved that when G = Dn acts on ordered adjacent vertices of G, the number of self-paired suborbits is n + 1 if n is odd and n + 2 if n is even. The study has also revealed a conjecture that gives a formula for computing the self-paired suborbits of the action of Dn on its ordered adjacent vertices. Pro-perties of suborbits are significant as they form a link between group theory and graph theory.