The Imprimitivity Action of the Product of Finite Alternating Groups on Cartesian Product of Ordered 𝜸� −Tuples of Finite Sets

Abstract

A group 𝐺 acts primitively on a set 𝑃 if the only 𝐺 -invariant partitions are trivial. This paper explores the primitivity of the product action of finite alternating groups 𝐴𝑛1 × 𝐴𝑛2 × … × 𝐴𝑛𝑚 on the Cartesian product of finite sets of ordered 𝛾-tuples. We demonstrate that for all 𝑛 − 𝛾 ≥ 2, this action is imprimitive by constructing explicit non-trivial block systems. This finding adds to previous research on the transitivity of such actions.