Hopf Galois structures on extensions of degree twice an odd prime square and their associated left braces

TL;DR

This paper classifies Hopf Galois structures and associated left braces on certain degree extensions, identifying when cyclic structures occur and counting their number.

Contribution

It provides a complete classification of Hopf Galois structures and left braces for extensions of degree twice an odd prime square, including cyclic cases and enumeration.

Findings

Classified all Hopf Galois structures for the specified extensions.

Determined which extensions admit cyclic Hopf Galois structures.

Counted the number of such structures for each extension.

Abstract

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.