Central series' and ($n$)-isoclinism of skew left braces
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arXiv
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| Format: | Preprint |
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2025
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| author | Kanrar, Arpan Roelants, Charlotte Yadav, Manoj K. |
| author_facet | Kanrar, Arpan Roelants, Charlotte Yadav, Manoj K. |
| contents | The aim of this article is to advance the knowledge on the theory of skew left braces. We introduce a subclass of skew left braces, which we denote by $\mathcal{I}_n$, $n \ge 1$, such that elements of the annihilator and lower central series' interact `nicely' with respect to commutation. That allows us to define a concept of $n$-isoclinism of skew left braces in $\mathcal{I}_n$, by using a concept of brace commutator words, which we have introduced. We prove results on $1$-isoclinism (isoclinism) of skew left braces analogous to important results in group theory. For any two symmetric $n$-isoclinic skew left braces $A$ and $B$, we prove that, there exist skew left braces $C$ and $R$ such that both $A$ and $B$ are $n$-isoclinic to both $C$ and $R$ and (i) $A$ and $B$ are quotient skew left braces of $C$; (ii) $A$ and $B$ are sub-skew left braces of $R$. Connections between a skew left brace and the group which occurs as a natural semi-direct product of additive and multiplicative groups of the skew left brace are investigated, and it is proved that $n$-isoclinism is preserved from braces to groups. We also show that various nilpotency concepts on skew left braces are invariant under $n$-isoclinism. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Central series' and ($n$)-isoclinism of skew left braces Kanrar, Arpan Roelants, Charlotte Yadav, Manoj K. Rings and Algebras 16T25, 81R50 The aim of this article is to advance the knowledge on the theory of skew left braces. We introduce a subclass of skew left braces, which we denote by $\mathcal{I}_n$, $n \ge 1$, such that elements of the annihilator and lower central series' interact `nicely' with respect to commutation. That allows us to define a concept of $n$-isoclinism of skew left braces in $\mathcal{I}_n$, by using a concept of brace commutator words, which we have introduced. We prove results on $1$-isoclinism (isoclinism) of skew left braces analogous to important results in group theory. For any two symmetric $n$-isoclinic skew left braces $A$ and $B$, we prove that, there exist skew left braces $C$ and $R$ such that both $A$ and $B$ are $n$-isoclinic to both $C$ and $R$ and (i) $A$ and $B$ are quotient skew left braces of $C$; (ii) $A$ and $B$ are sub-skew left braces of $R$. Connections between a skew left brace and the group which occurs as a natural semi-direct product of additive and multiplicative groups of the skew left brace are investigated, and it is proved that $n$-isoclinism is preserved from braces to groups. We also show that various nilpotency concepts on skew left braces are invariant under $n$-isoclinism. |
| title | Central series' and ($n$)-isoclinism of skew left braces |
| topic | Rings and Algebras 16T25, 81R50 |
| url | https://arxiv.org/abs/2503.10313 |