Automorphism Groups of a Family of Cayley Graphs of the Alternating Groups

Abstract

Let \(A_n\) be the alternating group of degree \(n\) with \(n \geq 5\). Set \(S = \{(1ij), (1ji) \mid 2 \leq i, j \leq n, i \neq j\}\). In this paper, it is shown that the full automorphism group of the Cayley graph \(\mathrm{Cay}(A_n, S)\) is the semi-product \(R(A_n) \rtimes \mathrm{Aut}(A_n, S)\), where \(R(A_n)\) is the right regular representation of \(A_n\) and \(\mathrm{Aut}(A_n, S) = \{\phi \in \mathrm{Aut}(A_n) \mid S^\phi = S\} \cong \mathrm{S_{n-1}}\).