# Cyclic complements and skew morphisms of groups

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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.