Classification of braces of cardinality p4

We classify nilpotent pre-Lie rings of cardinality p4 4 and thereby braces of the same cardinality, for a sufficiently large prime p. It has been shown that nilpotent pre-Lie rings of cardinality pncorrespond n correspond to strongly nilpotent braces of the same cardinality, for sufficiently large p. These braces are explicitly obtained from the corresponding pre-Lie rings by the construction of the group of flows. Braces of cardinality p4 4 that are not right nilpotent have already been classified. Consequently, our results complete the classification of braces with such cardinality. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).