Ranks and Subdegrees of the Action of the Product of Four Alternating Groups on the Cartesian Product of Four Sets of Ordered Pairs

Abstract

The rank and subdegrees of the action of 𝐴𝑛 × 𝐴𝑛 × 𝐴𝑛 × 𝐴𝑛 acting on 𝑊[2] × 𝑋 [2] × 𝑌 [2] × 𝑍 [2] are determined in this paper. Using combinatorics and mathematical induction, the rank of 𝐴𝑛 × 𝐴𝑛 × 𝐴𝑛 × 𝐴𝑛, ( ∀ 𝑛 ≥ 4) acting on 𝑊[2] × 𝑋 [2] × 𝑌 [2] × 𝑍 [2] , is 2 4 and the subdegrees are: 1, 4 (( 𝑛! (𝑛−2)! )− 1) , 6 (( 𝑛! (𝑛−2)! )− 1) 2 , 4 (( 𝑛! (𝑛−2)! )− 1) 3 and (( 𝑛! (𝑛−2)! )− 1) 4 .