From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures

The interplay between set-theoretic solutions of the Yang–Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf–Galois structures has spawned a considerable body of literature in recent years. In a recent paper, Alan Koch generalised a construction of Lindsay N. Childs, showing how one can obtain bi-skew braces (G,⋅,∘) from an endomorphism of a group (G,⋅) whose image is abelian. In this paper, we characterise the endomorphisms of a group (G,⋅) for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the set-theoretic solutions of the Yang–Baxter equation derived by Koch's construction carry over to our more general situation, and discuss the related Hopf–Galois structures.