On finite perfect two-sided skew braces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917519770517504 |
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| author | Damele, Marco |
| author_facet | Damele, Marco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22302 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On finite perfect two-sided skew braces Damele, Marco Group Theory 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. |
| title | On finite perfect two-sided skew braces |
| topic | Group Theory |
| url | https://arxiv.org/abs/2605.22302 |