Abstract
Let p be a prime number and let n be an integer not divisible by p and such that every group of order np has a normal subgroup of order p. (This holds in particular for p > n.) Under these hypotheses, we obtain a one-to-one correspondence between the isomorphism classes of braces of size np and the set of pairs (B-n, [t]), where Bn runs over the isomorphism classes of braces of size n and [t] runs over the classes of group morphisms from the multiplicative group of B-n to Z*(p) under a certain equivalence relation. This correspondence gives the classification of braces of size np from the one of braces of size n.
From this result we derive a formula giving the number of Hopf Galois structures of abelian type Z(p) x E on a Galois extension of degree np in terms of the number of Hopf Galois structures of abelian type E on a Galois extension of degree n. For a prime number p >= 7, we apply the obtained results to describe all left braces of size 12p and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree 12p. (c) 2023 The Author(s).
Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).