From pre-Lie rings back to braces

Abstract

Let A be a brace of cardinality p^{n} where p>n+1 is prime, and ann(p^{i}) be the set of elements of additive order at most p^{i} in this brace. A pre-Lie ring related to the brace A/ann(p^{2}) was constructed in [8]. We show that there is a formula dependent only on the additive group of the brace A which reverses the construction from [8]. As an application example it is shown that the brace A/ann(p^{4}) is the group of flows of a left nilpotent pre-Lie algebra.