Browsing by Author "Habineza, Olivier"

Now showing 1 - 1 of 1
  • Thumbnail Image
    Item
    On automorphism groups of the conjugacy class type Cayley graphs on the symmetric and alternating groups
    (Taylor and Francis Ltd., 2025) Habineza, Olivier; Mwambene, Eric
    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(An, S) on alternating groups An, 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 ∈ {An, Sn} 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 = An. 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(An, 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 = Sn, n > 4 and n ≠ 6, we show that Aut(Cay(Sn, S) ≅ (Hol(An) ⋊ 2) ≀ S2 for S ⊂ An \ {e}, and that Aut(Cay(Sn , S)) ≅ Hol(Sn) ⋊ 2 otherwise; provided that S does not contain Sn \ An or S ≠ An \ {e}, S ∉ {∅, Sn \ {e}}.