In this paper we determine the property of transitivity of the dihedral groups, , acting on , , and. We also compute the ranks and subdegrees of the respective actions.
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 S n 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 D n acting on ordered subsets of { } 1, 2, , X n = . The action of D n 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 D n 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 = D n 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 D n on its ordered adjacent vertices. Properties of suborbits are significant as they form a link between group theory and graph theory.
The study aims at determining the rank and subdegrees of the cyclic group, C n = <(12…n)> acting on X (r) , the set of unordered subsets of X={1, 2, …, n}. It has been shown that the action of C n on X (r) is transitive if and only ifr=1, r=n-1 orr=n. The rankfor transitive actions has been shown to ben. The number of self paired suborbits has been computed and conditions for paired suborbits discussed. The results have shown that the action of C n on X is equivalent to that of C n onX (n-1) .
Some properties of suborbital graphs corresponding to the action of on various sets have been studied. In this paper, we investigate some properties of suborbital graphs corresponding to the action of the Dihedral Groups, , on () .
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