On Grün's lemma for perfect skew braces
Cindy Tsang
TL;DR
This work investigates Grün's lemma analogues for skew braces by replacing the group center with the annihilator notions. It proves universal relations $\mathrm{Ann}_2(A)*(A*A)=1$ and $[\mathrm{Ann}_2(A),A*A]=1$ for any skew brace $A$, and shows $(A*A)*\mathrm{Ann}_2(A)=1$ for two-sided braces, with $\mathrm{Ann}(A/\mathrm{Ann}(A))=1$ when $A$ is two-sided and perfect. It then constructs explicit counterexamples via semidirect products $A=B\rtimes_\phi C$ with $B$ trivial and $C$ perfect, demonstrating that $(A*A)*\mathrm{Ann}_2(A)$ can fail and that $\mathrm{Ann}(A/\mathrm{Ann}(A))$ can be nontrivial even in perfect braces. Collectively, the results provide a genuine generalization of Grün's lemma to the skew-brace setting and clarify how $A*A$, $\mathrm{Ann}(A)$, and $\mathrm{Ann}_2(A)$ interact to determine center-like properties.
Abstract
By previous work of Cedó, Smoktunowicz, and Vendramin, one already knows that the analog of Grün's lemma fails to hold for perfect skew left braces when the socle is used as an analog of the center of a group. In this paper, we use the annihilator instead of the socle. We shall show that the analog of Grün's lemma holds for perfect two-sided skew braces but not in general.