Automorphism groups of Cayley graphs generated by connected transposition sets

Abstract

Let S be a set of transpositions that generates the symmetric group Sn, where n≥3. The transposition graph T(S) is defined to be the graph with vertex set {1,…,n} and with vertices i and j being adjacent in T(S) whenever (i,j)∈S. We prove that if the girth of the transposition graph T(S) is at least 5, then the automorphism group of the Cayley graph Cay(Sn,S) is the semidirect product R(Sn)⋊Aut(Sn,S), where Aut(Sn,S) is the set of automorphisms of Sn that fixes S. This strengthens a result of Feng on transposition graphs that are trees. We also prove that if the transposition graph T(S) is a 4-cycle, then the set of automorphisms of the Cayley graph Cay(S4,S) that fixes a vertex and each of its neighbors is isomorphic to the Klein 4-group and hence is nontrivial. We thus identify the existence of 4-cycles in the transposition graph as being an important factor in causing a potentially larger automorphism group of the Cayley graph.