On left nilpotent skew braces of class 2
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912370371067904 |
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| author | Ballester-Bolinches, A. Kurdachenko, L. A. Pérez-Calabuig, V. |
| author_facet | Ballester-Bolinches, A. Kurdachenko, L. A. Pérez-Calabuig, V. |
| contents | The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On left nilpotent skew braces of class 2 Ballester-Bolinches, A. Kurdachenko, L. A. Pérez-Calabuig, V. Group Theory Rings and Algebras 16T25, 16N40, 81R50 The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is best possible. |
| title | On left nilpotent skew braces of class 2 |
| topic | Group Theory Rings and Algebras 16T25, 16N40, 81R50 |
| url | https://arxiv.org/abs/2505.07115 |