Asian Journal of Pure and Applied Mathematics

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

Moses K. Maraka *

Department of Mathematics & Physical Sciences, Maasai Mara University; P.O. Box 861-20500, Narok, Kenya.

John W. Matuya

Department of Mathematics & Physical Sciences, Maasai Mara University; P.O. Box 861-20500, Narok, Kenya.

Edward M. Njuguna

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.

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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 A_n1×A_n2× …×A_nm on the Cartesian product of finite sets of ordered y -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.

Keywords: Transitivity action, primitivity action, ordered sets of tuples, cartesian product and alternating group

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