Graphs Whose Automorphism Groups Contain Or Represent The Alternating Groups, An And Symmetric Groups, Sn

ABSTRACT

There have been many investigations on the combinatorial structures and invariants over the group actions on the subsets of its elements. Studies on Group Theory have yielded varied and important results in the advancement of Algebra. Several studies have also been made on Graph Theory. Some Mathematicians have studied the concept of automorphisms on graphs thereby yielding important results. Automorphism groups from graphs containing the cyclic and dihedral groups, Cn and Dn respectively have been constructed using Schur’s Algorithm. In this project, we have extended the work to graphs whose Automorphism groups contain the Alternating Group An as well as those representing Symmetric group Sn. The graphs whose Automorphism groups contain or represent the Alternating Group An and Symmetric group Sn respectively have been constructed. Schur’s algorithm has been employed to construct these graphs. The actions of the Alternating Group An and the Symmetric group Sn have been shown to be transitive using the Cauchy- Frobenius Lemma and the Orbit-Stabilizer Theorem. The Automorphism graphs for the groups An and Sn have been constructed for 𝓃=3,4,𝐴𝑛𝑑 5.. The number of graphs whose groups of Automorphism contain An is 2 being the null and the complete graphs. The number of graphs whose groups of Automorphism represent Sn is 2 being the null and the complete graphs. We have presented the results of our findings from our workings as theorems and constructed the applicable graphs.