Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

Abstract

In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of <i>n</i> ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when <i>n</i> = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of <i>n</i> > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (<i>n</i> − 1).