The Cayley isomorphism property for Cayley maps

TL;DR

This paper investigates finite groups with the Cayley isomorphism property for Cayley maps, identifying restrictions on their structure and providing a classification and examples of such CIM-groups.

Contribution

It characterizes CIM-groups, offering a classification theorem and explicit examples of infinite series, advancing understanding of Cayley map symmetries.

Findings

CIM-groups have a very restricted structure

A short list of possible CIM-group candidates is provided

Infinite series of CIM-groups are explicitly constructed

Abstract

In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where a short list of possible candidates for CIM-groups is given. Theorem~\ref{111015c} provides concrete examples of infinite series of CIM-groups.