Brace blocks from bilinear maps and liftings of endomorphisms

On the connection between Hopf–Galois structures and skew braces

“…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 $$(G,\circ )$$.…”

“…Example Suppose that $$(G,\circ )$$ is nilpotent of class two, and define $$$$\begin{equation*} \sigma \cdot \tau =\sigma \;\circ\; {}^{\iota _{\circ }(\sigma )} \hspace{-1.111pt}\tau =\sigma \;\circ\; \sigma \;\circ\; \tau \;\circ\; \overline{\sigma }. \end{equation*}$$$$ Then by [20, Proposition 5.6], we have that $$(G,\cdot ,\circ )$$ is a bi‐skew brace and the gamma function of $$(G,\cdot ,\circ )$$ is given by $$\gamma (\sigma )=\iota _{\circ }(\overline{\sigma })$$. By Proposition 4.13, we derive that for the associated Hopf–Galois structure on $$L/K$$, the image of the Hopf–Galois correspondence consists precisely of the normal intermediate fields of $$L/K$$.…”

“…However, there is a way to obtain from this connection a bijective correspondence. Indeed, as a consequence of Theorem 2.3 (see [20,Section 7]), given a group (𝐺, •), there exists a bijective correspondence between group operations ⋅ such that (𝐺, ⋅, •) is a skew brace and regular subgroups of Perm(𝐺) normalised by 𝜆 • (𝐺), via…”

“…However, there is a way to obtain from this connection a bijective correspondence. Indeed, as a consequence of Theorem 2.3 (see [20, Section 7]), given a group $$(G,\circ )$$, there exists a bijective correspondence between group operations $$\cdot$$ such that $$(G,\cdot ,\circ )$$ is a skew brace and regular subgroups of $$\operatorname{Perm}(G)$$ normalised by $$\lambda _{\circ }(G)$$, via $$$$\begin{equation*} \cdot \mapsto \lambda _{\cdot }(G). \end{equation*}$$$$In this way, given a finite Galois extension of fields $$L/K$$ with Galois group $$(G,\circ )$$, we obtain a bijective correspondence between operations $$\cdot$$ such that $$(G,\cdot ,\circ )$$ is a skew brace and Hopf–Galois structure on $$L/K$$, which is a key observation for our new point of view.…”

“…Then by [20,Proposition 5.6], we have that (𝐺, ⋅, •) is a bi-skew brace and the gamma function of (𝐺, ⋅, •) is given by 𝛾(𝜎) = 𝜄 • (𝜎). By Proposition 4.13, we derive that for the associated Hopf-Galois structure on 𝐿∕𝐾, the image of the Hopf-Galois correspondence consists precisely of the normal intermediate fields of 𝐿∕𝐾.…”