A new formula for Lazard's correspondence for finite braces and pre-Lie algebras

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent F-p-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite F-p-braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent F-braces generated by one element of cardinality p(4) is obtained. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left nilpotent ideal, therefore every finite brace contains the largest left nilpotent ideal.(c) 2021 Elsevier Inc. All rights reserved.