Regular dessins uniquely determined by a nilpotent automorphism group

TL;DR

This paper classifies uniquely regular dessins with nilpotent automorphism groups, focusing on maximally automorphic p-groups of nilpotency class three, and reduces the problem to understanding their automorphism groups.

Contribution

It provides a classification of maximally automorphic p-groups of nilpotency class three, advancing the understanding of uniquely regular dessins with nilpotent automorphism groups.

Findings

Classification of maximally automorphic p-groups of nilpotency class three

Reduction of the problem to automorphism group analysis

Insights into uniquely regular dessins with nilpotent automorphism groups

Abstract

It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is reduced to the classification of finite maximally automorphic $p$-groups $G$, i.e., the order of the automorphism group of $G$ attains Hall's upper bound. Maximally automorphic $p$-groups of nilpotency class three are classified.