Automorphism group of the complete alternating group graph

摘要

Let Sn and An denote the symmetric group and alternating group of degree n with n ≥ 3, respectively. Let S be the set of all 3-cycles in Sn. The complete alternating group graph, denoted by CAGn, is defined as the Cayley graph Cay(An, S) on An with respect to S. In this paper, we show that CAGn (n ≥ 4) is not a normal Cayley graph. Furthermore, the automorphism group of CAGn for n ≥ 5 is obtained, which equals to Aut(CAGn)=(R(An)⋊Inn(Sn))⋊Z2≅(An⋊Sn)⋊Z2, where R(An) is the right regular representation of An, Inn(Sn) is the inner automorphism group of Sn, and Z2=〈h〉, where h is the map α↦α−1 (∀α ∈ An).