Powers of Skew-Morphisms

Skew-morphisms have important applications in the classification of regular Cayley maps, and have also been shown to be fundamental in the study of complementary products of finite groups AB with B cyclic and \(A\cap B = \{1\}\). As natural generalizations of group automorphisms, they share many of their properties but proved much harder to classify. Unlike automorphisms, not all powers of skew-morphisms are skew-morphisms again. We study and classify the powers of skew-morphisms that are either skew-morphisms or group automorphisms and consider reconstruction of skew-morphisms from such powers. We also introduce a new class of skew-morphisms that generalize the widely studied t-balanced skew-morphisms and which we call coset-preserving skew-morphisms. We show that, in certain cases, all skew-morphisms have powers that belong to this class and can therefore be reconstructed from these.

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