The Orbit Structure of the Product of Four Alternating Groups Acting on the Cartesian Product of Four Sets of Ordered Tuples

Abstract

This paper explores the orbit structure of the direct product of four alternating groups acting on the Cartesian product of four sets of ordered 𝛾-tuples. The associated orbitals and suborbits lengths are determined using combinatorial formulae. Contrary to previous studies involving ordered pairs or fewer group factors, this setting introduces significantly higher combinatorial configurations arising from higher-dimensional tuple structures. This brings rise to a general orbit structure pattern that do not arise in pair-based and fewer group factors cases. The number of orbitals of 𝐴𝑛1 × 𝐴𝑛2 × 𝐴𝑛3 × 𝐴𝑛4 acting on 𝑈1 [𝛾] × 𝑈2 [𝛾] × 𝑈3 [𝛾] × 𝑈4 [𝛾] , (∀ 𝑛 − 𝛾 ≥ 2) is 16 and explicit suborbits lengths formulae are obtained using combinatorial methods. The results extend existing research on ranks and subdegrees while revealing new structural phenomena unique to actions on ordered 𝛾-tuples of direct product of four alternating groups.