## Ranks, Subdegrees, Suborbital Graphs And Cycle Indice Sassociated With the Product Action Of An _ An _ An (4 _ N _ 8) On The Cartesian Product X _ Y _ Z

##### Abstract

Researchers over the years have studied group actions. The action of symmetric
and alternating groups on ordered and unordered subsets of a set X have been
considered leaving out the product action. In this study, we have considered the
product action of An An An on X Y Z, where n 4, X = f1; 2; 3; ; ng, Y =
fn+1; ; 2ng and Z = f2n+1; ; 3ng. This action is shown to be transitive using
the Orbit-Stabilizer theorem. The subdegrees and rank have been determined using
the de nition of an orbit of product action. The rank is 8 and the subdegrees are; 1,
(n1), (n1), (n1), (n 1)2; (n 1)2, (n 1)2 and (n 1)3. Sim's procedure has
been used in the construction of the corresponding non-trivial suborbital graphs. The
suborbital graphs of this action are undirected, regular and their girth is 3. All the
suborbital graphs except the one corresponding to the suborbit of length (n1)3, are
disconnected. The suborbital graphs corresponding to suborbits of length (n1) and
(n 1)2 have n2 and n components respectively. Finally, the cycle index formulas
of this action when n = 4; 5; 6; 7; 8, are derived using the product of monomials.