On automorphism groups of the conjugacy class type Cayley graphs on the symmetric and alternating groups

Abstract

The automorphism groups of Cayley graphs on symmetric groups, Cay(G, S), where S is a complete set of transpositions have been determined. In a similar spirit, automorphism groups of Cayley graphs Cay(A_n , S) on alternating groups A_n , where S is a set of all 3-cycles have also been determined. It has, in addition, been shown that these graphs are not normal. In all these Cayley graphs, one observes that their corresponding Cayley sets are a union of conjugacy classes. In this paper, we determine in their generality, the automorphism groups of Cay(G, S), where G ∈ {A_n , S_n } and S is a conjugacy class type Cayley set. Further, we show that the family of these graphs form a Boolean algebra. It is first shown that Aut(Cay(G, S)), S ∉ {∅, G \ {e}}, is primitive if and only if G = A_n . Using one of the results obtained by Praeger in 1990, we exploit further the other cases, thereby proving that, for n > 4 and n ≠ 6, Aut(Cay(A_n , S)) ≅ Hol(An ) ⋊ ₂, with Hol(G) ∼= G ⋊ Aut(G), provided that S is preserved by the outer automorphism defined by the conjugation by an odd permutation. Finally, in the remaining case G = S_n , n > 4 and n ≠ 6, we show that Aut(Cay(S_n, S) ≅ (Hol(A_n ) ⋊ ₂) ≀ S ₂ for S ⊂ A_n \ {e}, and that Aut(Cay(S_n , S)) ≅ Hol(S_n ) ⋊ ₂ otherwise; provided that S does not contain S_n \ A_n or S ≠ A_n \ {e}, S ∉ {∅, S_n \ {e}}.