On Gr\"{u}n's lemma for perfect skew braces
Group Theory
2026-04-27 v2 Quantum Algebra
Rings and Algebras
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.
Cite
@article{arxiv.2409.18410,
title = {On Gr\"{u}n's lemma for perfect skew braces},
author = {Cindy Tsang},
journal= {arXiv preprint arXiv:2409.18410},
year = {2026}
}
Comments
Slightly expanded Examples 3.6 and 3.7 as suggested by the referee. Also added Remark 3.8