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∩B={1}. A skew morphism of a group A is a bijection φ:A→A fixing the identity element of A and having the property that φ(xy)=φ(x)φπ(x)(y) for all x,y∈A, where π(x) depends only on x. The kernel of φ is the subgroup of all x∈A for which π(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 |A|, 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 Kovács 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.