Automorphism Groups of Cayley Maps

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“…Using Theorem 3, it follows that the automorphism groups of Cayley maps Aut(CM(G, X, p)) are rotary extensions of the underlying group G. This has been first observed in [6], where one can also find the following results relevant to the theory developed further in this section. Let M = CM(G, X, p) be a Cayley map, and let ρ be a bijection of the group G onto itself.…”

“…The concept of a rotary extension first occurred in relation to automorphism groups of Cayley maps in [6], where it was proved that the automorphism group Aut(M) of any Cayley map CM(G, X, p) is a rotary extension of the underlying group G by a group ρ generated by a special graph-automorphism ρ stabilizing the identity 1 G and called a "rotary mapping". Rotary extensions of groups form a special case of a much more general group extension discussed in [14].…”

“…Then k divides |X | and Aut(M) ∼ = G × rot ρ k , i.e., automorphism groups of Cayley maps are rotary extensions of the underlying group by a cyclic subgroup of order |X |/k. The paper [6] also provides us with a useful formula defining the rotary mapping ρ k : Let a be an arbitrary element of G, and let a = x 1 x 2 , . . .…”

“…In what follows, we shall use the above results from [6] to classify the finite groups isomorphic to some full automorphism group of a Cayley map.…”

“…Such a cyclic permutation of X clearly exists as X contains no involutions. Recall now that the results from [6] yield that Aut (CM(G, X, p)) ∼ = G if and only if the smallest divisor of |X | associated to a rotary mapping is |X | itself in which case the rotary mapping ρ |X | is simply equal to id G and Aut (CM(G, X, p)) is a rotary extension of G by a trivial group. We are going to alter the permutation p in such a way that will guarantee that none of the bijections ρ k defined by formula (3) and associated with a divisor k of |X |, k = |X |, will be equal to p k on X .…”