From braces to pre-Lie rings

TL;DR

This paper explores the relationship between braces and pre-Lie rings, constructing a pre-Lie ring from certain braces and analyzing their properties using powerful Lie rings, with applications to automorphisms and element commutativity.

Contribution

It introduces a novel construction linking braces to pre-Lie rings and studies their properties, especially for strongly nilpotent braces and powerful groups, expanding understanding of their algebraic structure.

Findings

Constructed a pre-Lie ring from braces modulo elements of order p^2.

Bounded the number of elements commuting with a given element in a brace.

Established a correspondence between powerful braces and powerful pre-Lie rings.

Abstract

Let $A$ be a brace of cardinality $p^{n}$ where $p>n+1$ is prime, and let $ann (p^{2})$ be the set of elements of additive order at most $p^{2}$ in this brace. We construct a pre-Lie ring related to the brace $A/ann(p^{2})$. In the case of strongly nilpotent braces of nilpotency index $k<p$ the brace $A/ann(p^{2})$ can be recovered by applying the construction of the group of flows to the resulting pre-Lie ring. We don't know whether, when applied to braces which are not right nilpotent, our construction is related to the group of flows. We use powerful Lie rings associated with finite $p$-groups in the study of brace automorphisms with few fixed points. As an application we bound the number of elements which commute with a given element in a brace, as well as the number of elements which multiplied from left by a given element give zero. We also study various Lie rings associated to powerful groups and braces whose adjoint groups are powerful, and show that the obtained Lie and pre-Lie rings are also powerful. We also show that braces whose adjoint groups are powerful and powerful left nilpotent pre-Lie rings are in one-to-one correspondence and that they are left and right nilpotent under some cardinality assumptions.