On Gr\"{u}n's lemma for perfect skew braces

TL;DR

This paper investigates the validity of Gr"{u}n's lemma in the context of perfect skew braces, demonstrating its applicability for perfect two-sided skew braces when using the annihilator.

Contribution

It introduces the use of the annihilator instead of the socle and establishes conditions under which Gr"{u}n's lemma holds for perfect skew braces.

Findings

Gr"{u}n's lemma holds for perfect two-sided skew braces with the annihilator.

The lemma does not hold in general for all perfect skew braces.

Using the annihilator instead of the socle is key to the positive result.

Abstract

By previous work of Ced\'{o}, Smoktunowicz, and Vendramin, one already knows that the analog of Gr\"{u}n'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 Gr\"{u}n's lemma holds for perfect two-sided skew braces but not in general.