The Orbit Structure of the Product of Four Alternating Groups Acting on the Cartesian Product of Four Sets of Ordered Tuples
Moses K. Maraka *
Department of Mathematics & Physical Sciences, Maasai Mara University, P.O. Box 861-20500, Narok, Kenya.
Lewis N. Nyaga
Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
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 y-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 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 -tuples of direct product of four alternating groups.
Keywords: Group action, cartesian product, rank and subdegrees