Almost all Cayley maps are mapical regular representations

TL;DR

This paper proves that as groups grow larger, almost all Cayley maps are mapical regular representations, meaning the original group coincides with their automorphism group.

Contribution

The paper establishes that the probability of a Cayley map being a mapical regular representation approaches 1 for large groups, extending previous deterministic results.

Findings

Proportion of MRRs among Cayley maps approaches 1 as group size increases.

Provides probabilistic analysis of Cayley maps and their automorphism groups.

Extends understanding of symmetry properties in large algebraic structures.

Abstract

Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.