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A permutation φ on a group G is called a skew morphism of G if φ (1) = 1 , and there exists an integer-valued function π : G → Z m , where m is the order of φ , such that φ (a b) = φ (a) φ π (a) (b) , for all a , b ∈ G . A skew morphism φ is smooth if the associated power function π of φ takes constant values on each orbit of φ . In this paper, we shall classify the smooth skew morphisms of semi-dihedral groups. [ABSTRACT FROM AUTHOR]

AbstractA finite group G is an exact product of two subgroups A and B if G = AB and A∩B={1G}. In this paper, for all odd numbers k≥3, using the theory of skew morphisms and associated extended power functions, we present a classification of exact products of a cyclic group and a dihedral group D2k. As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified. [ABSTRACT FROM AUTHOR]

A skew morphism of a finite group A is a permutation φ on A fixing the identity element of A, and for which there exists an integer-valued function π : A → Z | φ | on A such that φ (a b) = φ (a) φ π (a) (b) for all a , b ∈ A . Moreover, the period of φ is the smallest positive integer d such that π (φ d (a)) ≡ π (a) (mod | φ |) for all a ∈ A . In the case where d = 1 , the skew morphism φ is called smooth. It is well known that if φ is a skew morphism of period d, then φ d is a smooth skew morphism. Thus, every skew morphism of period d may be extracted as a dth root of a smooth skew morphism. In this paper, we introduce a new concept of average function to investigate skew morphisms and as an application we present a classification of smooth skew morphisms of the dicyclic groups. [ABSTRACT FROM AUTHOR]

We use the term half-regular map to describe an orientable map with an orientation preserving automorphism group that is transitive on vertices and half-transitive on darts. We present a full classification of half-regular Cayley maps using the concept of skew-morphisms. We argue that half-regular Cayley maps come in two types: those that arise from two skew-morphism orbits of equal size that are both closed under inverses and those that arise from two equal-sized orbits that do not contain involutions or inverses but one contains the inverses of the other. In addition, half-regular Cayley maps of the first type are shown to be half-edge-transitive, while half-regular Cayley maps of the second type are shown to be necessarily edge-transitive. A connection between half-regular Cayley maps and regular hypermaps is also investigated. [ABSTRACT FROM AUTHOR]