ON COMBINATORIAL PROPERTIES, INVARIANTS AND STRUCTURES OF THE ACTION OF An X An ON X x Y

Abstract

The concept of combinatorial properties, invariants, structures as well as the pairing of suborbits associated with the action of the actions of the direct product , of the alternating group by the alternating group alternating on the Cartesian product where and are disjoint sets each containing elements has never attention from researchers for a very long time. This paper is concerned with proving that the action is both transitive and imprimitive when Also, it shall establish that the rank is 9 whenever but is 4 for all In addition, it shows that the subdegrees associated with the action are Lastly, it is shown that all the suborbits corresponding to the action, are self-paired when n n A A  nA n A , Y X    n x x x x X , , , , 3 2 1     n y y y y Y , , , , 3 2 1   n . 3n , 3  n . 4n     . 1, 1, 1, 12    n n n .