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

Abstract

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 8 and the subdegrees of 𝐴𝑛1 × 𝐴𝑛2 × 𝐴𝑛3 , on 𝑃1 [𝛾] × 𝑃2 [𝛾] × 𝑃3 [𝛾] are: 1, (( 𝑛! (𝑛−𝛾)! )−1) , (( 𝑛! (𝑛−𝛾)! )− 1) , (( 𝑛! (𝑛−𝛾)! )− 1) , (( 𝑛! (𝑛−𝛾)! )− 1) 2 , (( 𝑛! (𝑛−𝛾)! )− 1) 2 , (( 𝑛! (𝑛−𝛾)! )−1) 2 and (( 𝑛! (𝑛−𝛾)! )−1) 3 .