Commutator-central maps, brace blocks, and {H}opf-{G}alois structures on {G}alois extensions

TL;DR

This paper introduces a new method for constructing bi-skew braces called brace blocks using compatible endomorphisms of nonabelian groups, and explores their relation to Hopf-Galois structures, especially for groups of nilpotency class two.

Contribution

It presents a novel construction of brace blocks from compatible endomorphisms and links these to Hopf-Galois structures on Galois extensions, with detailed examples.

Findings

Constructed brace blocks from compatible endomorphisms of nonabelian groups.

Connected brace block constructions to Hopf-Galois structures.

Identified the maximal brace block for the quaternion group.

Abstract

Let $G$ be a nonabelian group. We show how a collection of compatible endomorphisms $\psi_i:G\to G$ such that $\psi_i([G,G])\le Z(G)$ for all $i$ allows us to construct a family of bi-skew braces called a brace block. We relate this construction to other brace block constructions and interpret our results in terms of Hopf-Galois structures on Galois extensions. We give special consideration to the case where $G$ is of nilpotency class two, and we provide several examples, including finding the maximal brace block containing the group of quaternions.