The Cayley Isomorphism Property for Cayley Maps

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.