On automorphism groups of the conjugacy class type Cayley graphs on the symmetric and alternating groups
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Date
2025
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Publisher
Taylor and Francis Ltd.
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(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}}.
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Keywords
Boolean Algebra of Cayley Graphs, Primitive Groups, Graph Covers, Conjugacy Classes for Groups, Automorphism Groups
Citation
Habineza, O. and Mwambene, E., 2025. On automorphism groups of the conjugacy class type Cayley graphs on the symmetric and alternating groups. Quaestiones Mathematicae, pp.1-19.