Skew-morphisms of regular Cayley maps

Regular Cayley maps for finite abelian groups

“…A bijection f of the group A satisfying both this equation and f (1) = 1 is called a skew morphism [12] for the group A, with associated power function p. Note that a skew morphism of A is a group automorphism if and only if its power function is identically 1 on a generating set for A.…”

Restricted skew-morphisms on matrix algebras

“…Maps satisfying (1) will be called restricted skew-morphisms to distinguish them from skew-morphisms. Skew-morphisms were recently introduced by Jajcay andŠiráň [6] as bijective unital maps on groups with property (1) but where κ(A) takes the values in Z, the set of all integers. It needs to be said that in [6] they considered only unital bijections on finite groups, which are consequently of finite order and allow one to replace negative powers of φ by nonnegative powers, modulo the order of φ. Jajcay andŠiráň used skewmorphisms in an attempt to give a unified treatment of regular Cayley maps, which by definition are 2-cell embeddings of Cayley graphs into orientable surfaces which preserve a given orientation at each vertex.…”

“…We also prove that a surjective restricted skewmorphism has a finite order if its power function takes a value which is greater than one on GL n , see Theorem 18. The case when the power function takes the value zero on GL n was treated in [6], see also Lemma 1 below.…”

A Classification of Prime-Valent Regular Cayley Maps on Abelian, Dihedral and Dicyclic Groups

“…It is easy to see that for every g ∈ G, L g R = RL g , henceG is a subgroup of Aut (M) acting regularly on vertices. Furthermore, a Cayley map M = CM (G, X, q) is regular if and only if there exists an automorphism ρ in the stabilizer (Aut (M)) v of a vertex v cyclically permuting the |X| arcs based at v. In this case, Aut (M) is a product ofG with a cyclic group ρ ∼ = Z n , where n = |X| (see [4,5]). …”

“…On the other hand, a Cayley map M is anti-balanced if M is (−1)-balanced [13]. For more general theory of Cayley maps and their automorphisms, the reader is referred to [4,5].…”