Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all <i>n</i> ≥ 2 and the suborbits are length 1, (<i>n</i>-1), (<i>n</i>-1), (<i>n</i>-1), (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>3</sup>. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all <i>n</i> > 2 and suborbital graphs of the group action are undirected. It is shown that graphs <I>Γ<sub>2</sub></I> and <I>Γ</I><sub>3</sub> are regular of degree <i>n</i>-1, graphs <I>Γ</I><sub>4</sub>, <I>Γ</I><sub>5</sub> and <I>Γ</I><sub>6</sub> of degree (<i>n</i>-1)<sup>2</sup> and graph <I>Γ</I><sub>7</sub> is regular of degree (<i>n</i>-1)<sup>3</sup>. The suborbital graphs <i>Γ<sub>i</sub></i>(<i>i=</i>1, 2,…, 6) are disconnected, with the number of connected components equal to <i>n</i><sup>2</sup> while suborbital graph <I>Γ</I><sub>7</sub><i> </i>is connected for all <i>n</i> > 2.