On Combinatorial Properties, Invariants and Structures of the Action of An×An on X×Y

Abstract

This paper investigates the permutation group action of the direct product A_n×A_n on the Cartesian product X×Y, where X=(x₁,x₂,.....x_n) and Y=(y₁,y₂,.....y_n) are disjoint sets of equal cardinality. Despite the natural symmetry of this action, its combinatorial and structural properties have received little attention in the literature. We prove that the action is transitive and imprimitive for all n is greater than or equal to 3. The rank of the action is shown to be 9 for n=3 and subdegrees 1^×9.  For all n is greater than or equal to 4, the subdegrees are 1, (n-2)^×2, (n-1)². Furthermore, we demonstrate that all  nontrivial suborbits are self-paired when n is greater than or equal to 4, and we construct the corresponding suborbital graphs, analyzing their regularity, connectedness, and girth.