AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 51 (2011), Pages 157–173 On automorphisms of Cayley graphs and irregular groups

Let G be a group and CayP (G) < Sym(G) be the subgroup of all permutations that induce graph automorphisms on every Cayley graph of G. The group G is graphically abelian if the map ν: g → g −1 belongs to CayP (G); these groups have been classified. Also G is irregular if there exists σ ∈ CayP (G) such that σ = 1G, σ(1) = 1 and σ = ν. We show G is irregular if and only if G =Dic(A, I); every non-abelian graphically abelian group is irregular; and if G is irregular but not graphically abelian, σ ∈ CayP (G) andσ(1) = 1, then σ ∈ Aut(G). No irregular group has a GRR. If an irregular group G is not graphically abelian then there is exactly one irregular map σ and CayP (G) ∼ = G⋊〈σ〉, orotherwise CayP (G) ∼ = (G ⋊ Inn(G)) ⋊ 〈ν〉. 1 Basic Definitions A non-empty subset S of a group G is called a Cayley subset of G provided 1 / ∈ S and for all s ∈ G, ifs ∈ S then s −1 ∈ S. The Cayley graph, Cay(G, S), may b