# Hopf-Galois structures on a Galois $S_n$-extension

## TL;DR

This paper determines the exact number of Hopf-Galois structures on Galois extensions with symmetric group Galois groups, providing precise enumeration for these algebraic structures.

## Contribution

It offers a complete enumeration of Hopf-Galois structures specifically for Galois extensions with symmetric group Galois groups, a case not fully understood before.

## Key findings

- Exact count of Hopf-Galois structures for Galois $S_n$-extensions
- Provides formulas or methods for enumeration
- Enhances understanding of algebraic structures in Galois theory

## Abstract

In this paper, we shall determine the exact number of Hopf-Galois structures on a Galois $S_n$-extension, where $S_n$ denotes the symmetric group on $n$ letters.