On the connection between Hopf--Galois structures and skew braces

Abstract

We present a different version of the well-known connection between Hopf--Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman. We show that the known results that involve this connection easily carry over to this new perspective, and that new ones naturally appear. As an application, we present new insights on the study of the surjectivity of the Hopf--Galois correspondence, explaining in more detail the role of bi-skew braces in Hopf--Galois theory.

arXiv commentPublished in the Bulletin of the London Mathematical Society

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• On two-sided skew braces
• A classification of module braces over the ring of $\mathbf{p}$-adic integers
• Classification of the types for which every Hopf--Galois correspondence is bijective