Brace blocks from bilinear maps and liftings of endomorphisms

TL;DR

This paper develops new methods to construct brace blocks using bilinear maps and liftings of endomorphisms, providing examples that include infinite sequences converging to a given group operation.

Contribution

It extends Koch's constructions of brace blocks by introducing techniques involving bilinear maps and endomorphism liftings, with new examples including an answer to Greither's question.

Findings

Constructed brace blocks with arbitrary cardinality.

Provided examples including an infinite sequence converging to a group operation.

Answered a question of Greither with a novel example.

Abstract

We extend two constructions of Alan Koch, exhibiting methods to construct brace blocks, that is, families of group operations on a set $G$ such that any two of them induce a skew brace structure on $G$. We construct these operations by using bilinear maps and liftings of endomorphisms of quotient groups with respect to a central subgroup. We provide several examples of the construction, showing that there are brace blocks which consist of distinct operations of any given cardinality. One of the examples we give yields an answer to a question of Cornelius Greither. This example exhibits a sequence of distinct operations on the $p$-adic Heisenberg group $(G, \cdot)$ such that any two operations give a skew brace structure on $G$ and the sequence of operations converges to the original operation "$\cdot$".