On finite perfect two-sided skew braces

TL;DR

This paper establishes a classification and structural decomposition for finite perfect two-sided skew braces, revealing their relation to perfect groups and extending existing classifications.

Contribution

It introduces a central product theory for skew braces and proves a canonical decomposition for finite perfect two-sided skew braces.

Findings

Finite perfect two-sided skew braces decompose into almost trivial and trivial parts.

Perfectness of the skew brace is equivalent to perfectness of its additive or multiplicative group.

Classification extends to trivial-center cases, recovering known classifications.

Abstract

We prove a structure theorem for finite perfect two-sided skew braces. The main tool is a central product theory for skew braces, developed here in both external and internal form; we show that these two constructions are equivalent. Our main result states that every finite perfect two-sided skew brace \(B\) admits the canonical decomposition $B=B^2\circ B^{2,\operatorname{op}},$ where \(B^2\) is almost trivial with perfect additive group, while \(B^{2,\operatorname{op}}\) is trivial with perfect additive group. Thus finite perfect two-sided skew braces are classified, up to central amalgamation, by trivial and almost trivial skew braces arising from perfect groups. This decomposition has strong consequences for the underlying groups: for finite two-sided skew braces, perfectness of the skew brace is equivalent to perfectness of either the additive or the multiplicative group. In the trivial-center case the central product becomes a direct product, recovering Trappeniers' classification of finite simple two-sided skew braces. We also show that quasi-simple two-sided skew braces are necessarily either trivial or almost trivial. Finally, we prove that this rigidity is genuinely two-sided by constructing a quasi-simple skew brace which is not two-sided and is neither trivial nor almost trivial.