Abstract
Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality pn, for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality pn for n+1<p. As an application, by using well known results about the correspondence between braces and Hopf-Galois extensions we use pre-Lie algebras to describe Hopf-Galois extensions.
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