From braces to pre-Lie rings

Abstract

Let A A be a brace of cardinality p n p^{n} where p > n + 1 p>n+1 is prime and let a n n ( p 2 ) ann (p^{2}) be the set of elements of additive order at most p 2 p^{2} in this brace. We construct a pre-Lie ring related to the brace A / a n n ( p 2 ) A/ann(p^{2}) . In the case of strongly nilpotent braces of nilpotency index k > p k>p the brace A / a n n ( p 2 ) A/ann(p^{2}) can be recovered by applying the construction of the group of flows to the resulting pre-Lie ring. We do not know whether or not our construction is related to the group of flows when applied to braces which are not right nilpotent.