Skew brace extensions, second cohomology and complements

Abstract

We study extensions and second cohomology of skew left braces via the natural semi-direct products associated with the skew left braces. Let $0 \to I \to E \to H \to 0$ be a skew brace extension and $\Lambda_H$ denote the natural semi-direct products associated with the skew left brace $H$. We establish a group homomorphism from ${\rm H}_{Sb}^2(H, I)$ into ${\rm H}_{Gp}^2(\Lambda_H, I \times I)$, which turns out to be an embedding when $I \le {\rm Soc}(E)$. In particular the Schur multiplier of a skew left braces $H$ embeds into the Schur multiplier of the group $\Lambda_H$. Analog of the Schur-Zassenhaus theorem is established for skew left braces in several specific cases. We introduce a concept called minimal extensions (which stay at the extreme end of split extensions) of skew left braces and derive many fundamental results. Several reduction results for split extensions of finite skew left braces by abelian groups (viewed as trivial left braces) are obtained.