On bi-skew braces and brace blocks

On the connection between Hopf–Galois structures and skew braces

“…By [10, table at page 1175], a skew brace is a bi‐skew brace if and only if its gamma function has values in $$\operatorname{Aut}(G,\circ )$$. If $$(G,\cdot ,\circ )$$ is a bi‐skew brace, then the left ideals of $$(G,\cdot ,\circ )$$ and $$(G,\circ ,\cdot )$$ coincide; see [38, Lemma 3.1].…”

“…For example, in [7], a key role for the study of the Hopf–Galois module structure of a Galois extension $$L/K$$ of $$p$$‐adic fields of degree $$p^2$$, $$p$$ a prime, was played by an intermediate normal field $$F$$ of $$L/K$$ such that, given a Hopf–Galois structure on $$L/K$$, $$F$$ is in the image of the Hopf–Galois correspondence, and the Hopf–Galois structure on $$L/K$$ yields the classical structures on $$L/F$$ and $$F/K$$. More generally, all the skew braces obtained with [38, Theorem 6.6], which generalise several constructions developed in recent years, are metatrivial, so similar reasonings could be repeated.…”

“…We deal now with bi‐skew braces $$(G,\cdot ,\circ )$$ whose gamma functions have values in the inner automorphism group of $$(G,\circ )$$; these skew braces have been recently studied in [19, 20, 27, 28, 38]. Denote by $$Z(G)$$ the centre of $$(G,\circ )$$ and by $$N(G)$$ the norm of $$(G,\circ )$$, that is, the intersection of the normalisers of the subgroups of $$(G,\circ )$$.…”

“…Proof Let $$\psi \colon (G,\circ )\rightarrow N(G)/Z(G)$$ be a group homomorphism. By the main theorem of [37], the quotient $$N(G)/Z(G)$$ is abelian, so we can apply [38, Theorem 6.6] to derive that $$(G,\cdot _{\psi },\circ )$$ is a bi‐skew brace and the gamma function of $$(G,\cdot _{\psi },\circ )$$ is given by $$\gamma (\sigma )=\iota _{\circ }(\overline{\psi (\sigma )})$$. In particular, the gamma function of $$(G,\cdot _{\psi },\circ )$$ is given by conjugation by elements of $$N(G)$$ in $$(G,\circ )$$, so by power automorphisms of $$(G,\circ )$$, and therefore we obtain our assertion.$$\Box$$…”

“…We deal now with bi-skew braces (𝐺, ⋅, •) whose gamma functions have values in the inner automorphism group of (𝐺, •); these skew braces have been recently studied in [19,20,27,28,38]. Denote by 𝑍(𝐺) the centre of (𝐺, •) and by 𝑁(𝐺) the norm of (𝐺, •), that is, the intersection of the normalisers of the subgroups of (𝐺, •).…”