On Grün's lemma for perfect skew braces

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.

arXiv commentSlightly expanded Examples 3.6 and 3.7 as suggested by the referee. Also added Remark 3.8

References in corpus (8)

• Skew braces and the Yang-Baxter equation
• On skew braces (with an appendix by N. Byott and L. Vendramin)
• Skew left braces of nilpotent type
• Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
• Isoclinism of skew braces
• Schur covers of skew braces
• On two-sided skew braces
• A generalization of Ito's theorem to skew braces