Cyclic complements and skew morphisms of groups

A skew morphism of a group is a generalisation of an automorphism, arising in the context of regular Cayley maps or of groups expressible as a product AB of subgroups A and B with B cyclic and A boolean AND B = {1}. A skew morphism of a group A is a bijection phi : A -> A fixing the identity element of A and having the property that phi(xy) = phi(x)phi(pi(x)) (y) for all x, y is an element of A, where pi(x) depends only on x. The kernel of phi is the subgroup of all x is an element of A for which pi(x) = 1. In this paper, we present a number of previously unknown properties of skew morphisms, one being that if A is any finite group, then the order of every skew morphism of A is less than vertical bar A vertical bar, and another being that the kernel of every skew morphism of a non-trivial finite group is non-trivial. We also prove a number of theorems about skew morphisms of finite abelian groups, some of which simplify or extend recent theorems of Kovacs and Nedela [13]. For example, we determine all skew morphisms of the finite abelian groups whose order is prime, or the square of a prime, or the product of two distinct primes. In addition, we completely determine the finite abelian groups for which every skew morphism is an automorphism. (C) 2016 Elsevier Inc. All rights reserved.