Transitivity Action of the Cartesian Product of the Alternating Group Acting on a Cartesian Product of Ordered Sets of Triples
Maraka K. Moses *
Department of Physical Sciences, Chuka University; P.O. Box 109-60400 Chuka, Kenya.
Musundi W. Sammy *
Department of Physical Sciences, Chuka University; P.O. Box 109-60400 Chuka, 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
In this paper, we investigate some transitivity action properties of the cartesian product of the alternating group \(A_{n}(n \geq 5)\) acting on a cartesian product of ordered sets of triples using the Orbit-Stabilizer Theorem by showing that the length of the orbit \((p, s, v) \text { in } A_{n} \times A_{n} \times A_{n},(n \geq 5)\) acting on \(P^{[3]} \times S^{[3]} \times V^{[3]}\) is equivalent to the cardinality of \(P^{[3]} \times S^{[3]} \times V^{[3]}\) to imply transitivity.
Keywords: Orbits, stabilizer, transitivity action, ordered sets of triples, cartesian product, fixed point