The automorphism group of the alternating group graph

Abstract

Let A n be the alternating group of degree n with n ≥ 3 . Set S = { ( 1 2 i ) , ( 1 i 2 ) ∣ 3 ≤ i ≤ n } . The alternating group graph, denoted by A G n , is defined as the Cayley graph on A n with respect to S . Jwo et al. (1993) [J.-S. Jwo, S. Lakshmivarahan, S.K. Dhall, A new class of interconnection networks based on the alternating group, Networks 23 (1993) 315–326] introduced the alternating group graph A G n as an interconnection network topology for computing systems, and they proved that A G n is arc-transitive. In this work, it is shown that the full automorphism group of A G n is the semi-direct product R ( A n ) ⋊ Aut ( A n , S ) , where R ( A n ) is the right regular representation of A n and Aut ( A n , S ) = { α ∈ Aut ( A n ) ∣ S α = S } ≅ S n − 2 × S 2 . It follows from this result that A G n is arc-transitive but not 2-arc-transitive.