Counterexample to a conjecture about braces

TL;DR

This paper presents a counterexample of a finite solvable group, specifically a p-group, that does not admit a left brace structure, challenging a previous conjecture in brace theory.

Contribution

It provides the first known example of a finite solvable group without a left brace structure, advancing understanding in the algebraic theory of braces.

Findings

Identifies a finite p-group lacking a left brace structure

Improves upon Rump's argument using cross-disciplinary methods

Relates brace theory to other algebraic areas

Abstract

We find an example of a finite solvable group (in fact, a finite $p$-group) without any left brace structure (equiv. which is not an IYB group). Our argument is an improvement of an argument of Rump, using previous work in other areas of Burde, and of Featherstonhaugh, Caranti and Childs, which we relate to brace theory.