Inducing braces and Hopf Galois structures

Abstract

Let $p$ 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 $12 p$.