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

Abstract

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite Fp-braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent Fp-braces generated by one element of cardinality p4 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.