GRAPHS WHOSE AUTOMORPHISM GROUPS CONTAIN THE ALTERNATING GROUPS

There have been many investigations on the combinatorial structures and invariants over the group actions on the subsets of its elements. Automorphism groups from Graphs containing the cyclic and dihedral groups, C_n and D_n respectively have been constructed using Schur’s Algorithm. In this paper, we seek to extend the work to graphs whose automorphism groups contain the Alternating group A_n. We plan to construct the graphs whose automorphism groups contain the Alternating group A_n. To construct these graphs, we will engage Schur’s algorithm. We will first consider cases for , which will in turn help us make generalizations for any value of, the degree of the group in study. The process will involve, establishing the transitivity of the Alternating group A_n first, finding the adjacency matrices and then constructing the said graphs. We then determine a formula for calculating the number of graphs whose automorphism groups contain the Alternating group A_n. Through this study we have yielded important concepts and results in the field of group theory. We present the results of our findings from our workings as lemmas, graphs, and or tables where applicable.