### The Cayley Isomorphism Property for Cayley Maps

#### Abstract

The* Cayley Isomorphism* property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, then the map is an embedding of a Cayley graph into an oriented surface with the same cyclic rotation around every vertex.

Two Cayley maps are called *Cayley isomorphic* if there exists a map isomorphism between them which is a group isomorphism too. We say that a finite group $H$ is a *CIM-group* if any two Cayley maps over $H$ are isomorphic if and only if they are Cayley isomorphic.

The paper contains two main results regarding CIM-groups. The first one provides necessary conditons for being a CIM-group. It shows that a CIM-group should be one of the following

$$

\mathbb{Z}_m\times\mathbb{Z}_2^r, \

\mathbb{Z}_m\times\mathbb{Z}_{4},\

\mathbb{Z}_m\times\mathbb{Z}_{8}, \ \mathbb{Z}_m\times Q_8, \

\mathbb{Z}_m\rtimes\mathbb{Z}_{2^e}, e=1,2,3,$$ where $m$ is an odd square-free number and $r$ a non-negative integer. Our second main result shows that the groups $\mathbb{Z}_m\times\mathbb{Z}_2^r$, $\mathbb{Z}_m\times\mathbb{Z}_{4}$, $\mathbb{Z}_m\times Q_8$ contained in the above list are indeed CIM-groups.