Inducing braces and Hopf Galois structures

TL;DR

This paper characterizes the structure of left braces of size np, where p is prime and n is not divisible by p, and derives formulas for counting Hopf Galois structures of certain abelian types on degree np extensions.

Contribution

It provides a method to construct all braces of size np from smaller braces and classifies Hopf Galois structures of abelian type on degree np extensions.

Findings

Left braces of size np can be obtained as semidirect products.

Derived a formula for counting Hopf Galois structures of abelian type.

Explicitly described braces of size 12p and counted associated 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 $np$ has a normal subgroup of order $p$. (This holds in particular for $p>n$.) We prove that left braces of size $np$ may be obtained as a semidirect product of the unique left brace of size $p$ and a left brace of size $n$. We give a method to determine all braces of size $np$ from the braces of size $n$ and certain classes of morphisms from the multiplicative group of these braces of size $n$ to $\mathrm{Z}_p^*$. From it we derive a formula giving the number of Hopf Galois structures of abelian type $\mathrm{Z}_p \times E$ 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 $p\geq 7$, 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 $12p$.