Ranks and Subdegrees of the Action of the Product of Three Alternating Groups on the Cartesian Product of Three Sets of Ordered γ-Tuples

In this paper, the ranks and subdegrees of the action of the product of three alternating groups, An1×An2×An3, acting on the Cartesian product of three sets of ordered γ –tuples, P1[γ] × P2[γ] × P3[γ], are determined. Using combinatorial formula and mathematical induction, ∀ n - γ ≥ 2, the rank of An1 × An2 × An3 acting on P1[γ] × P2[γ] × P3[γ] is and the subdegrees of An1 × An2 × An3, on P1[γ] × P2[γ] × P3[γ] are: 1,