Authors: Salihu Aliyu Lawan, Shuaibu Garba Ngulde, Babagana Ibrahim Bukar
Abstract: In this paper, we study some action of S6 on A6. With Particular cases for n 2, 3, …,. and provide new combinatorial and structural insight into direct product actions of symmetric groups. Groups, Algorithms and Programming software (GAP) have been used to compute the elements of stabilizer S6. Orbit-stabiliser theorem and Cauchy-Frobenius lemma were applied to determine the number of S6(x)-orbits and their corresponding length respectively. We established that the action is transitive, faithful and imprimitive for n ≥ 2. Further results include explicit descriptions of point stabilizers, computation of orbit sizes using the Orbit–Stabilizer Theorem. We also established kernel of the action S6 on A6, and the construction of associated suborbitalni graphs and Upper bounds for the diameter of the resulting graphs are obtained, we the generalized the action (Sn)k for any n ≥ 2 on Cartesian products of k sets.
