Automorphisms of the Alternating Group Network

Abstract

Let \(A_n\) be the alternating group of degree \(n\) with \(n > 4\). Set \(T = \{(1 2 3), (1 3 2), (1 2)(3 i) \mid 4 \leq i \leq n\}\). The alternating group network, denoted by \(AN_n\), is defined as the Cayley graph on \(A_n\) with respect to \(T\). Some properties of \(AN_n\) have been investigated in [App. Math.—JCU, Ser. A 14 (1998) 235-239; IEEE Trans. Comput. 55 (2006) 1645-1648; Inform. Process. Lett. 110 (2010) 403-409; J. Supercomput. 54 (2010) 206-228]. In this paper, it is shown that the full automorphism group of \(AN_n\) is the semi-direct product \(R(A_n) \rtimes \text{Aut}(A_n, T)\), where \(R(A_n)\) is the right regular representation of \(A_n\) and \(\text{Aut}(A_n, T) = \{\alpha \in \text{Aut}(A_n) \mid T^\alpha = T\} \cong S_{n-3} \times S_2\).