# Cyclic complements and skew morphisms of groups

### Marston Conder University of Auckland

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Minisymposium: GRAPH IMBEDDINGS AND MAP SYMMETRIES

Content: A skew morphism of a group is a variant of an automorphism, which arises in the study of regular Cayley maps (regular embeddings of Cayley graphs on surfaces, with the property that the ambient group induces a vertex-regular group of automorphisms of the embedding). More generally, skew morphisms arise in the context of any group expressible as a product $AB$ of subgroups $A$ and $B$ with $B$ cyclic and $A\cap B = \{1\}$. Specifically, a skew morphism of a group $A$ is a bijection $\varphi\!: A \to A$ fixing the identity element of $A$ and having the property that $\varphi(xy) = \varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y \in A$, where $\pi(x)$ depends only on $x$. The kernel of $\varphi$ is the subgroup of all $x \in A$ for which $\pi(x) = 1$. In this talk I will present some of the theory of skew morphisms, including some new theorems: two about the order and kernel of a skew morphism of a finite group, and a complete determination of the finite abelian groups for which every skew morphism is an automorphism. Much of this is joint work with Robert Jajcay and Tom Tucker.

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