Crespo Vicente, TeresaGil Muñoz, DanielRio, AnnaVela del Olmo, Ma. Montserrat (Maria Montserrat)2025-01-202025-01-202023-09-010022-4049https://hdl.handle.net/2445/217659Let $p$ be a prime number and let $n$ be an integer not divisible by $p$ and such that every group of order $n p$ 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 $n p$ and the set of pairs $\left(B_n,[\tau]\right)$, where $B_n$ runs over the isomorphism classes of braces of size $n$ and $[\tau]$ runs over the classes of group morphisms from the multiplicative group of $B_n$ to $\mathbf{Z}_p^*$ under a certain equivalence relation. This correspondence gives the classification of braces of size $n p$ 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 $\mathbf{Z}_p \times E$ on a Galois extension of degree $n p$ in terms of the number of Hopf Galois structures of abelian type $E$ on a Galois extension of degree $n$. For a prime number $p \geq 7$, we apply the obtained results to describe all left braces of size $12 p$ and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree $12 p$.16 p.application/pdfengcc by-nc-nd (c) Teresa Crespo Vicente et al., 2023http://creativecommons.org/licenses/by-nc-nd/3.0/es/Àlgebres de HopfGrups de permutacionsExtensions de cossos (Matemàtica)Hopf algebrasPermutation groupsField extensions (Mathematics)info:eu-repo/semantics/article7516712025-01-20info:eu-repo/semantics/openAccess