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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {I}_n\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n \ge 1\)</EquationSource> </InlineEquation>, such that elements of the annihilator and lower central series interact ‘nicely’ with respect to commutation. That allows us to define a concept of <i>n</i>-isoclinism of skew left braces in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {I}_n\)</EquationSource> </InlineEquation>, 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 <i>n</i>-isoclinic skew left braces <i>A</i> and <i>B</i>, we prove that, there exist skew left braces <i>C</i> and <i>R</i> such that both <i>A</i> and <i>B</i> are <i>n</i>-isoclinic to both <i>C</i> and <i>R</i> and (i) <i>A</i> and <i>B</i> are quotient skew left braces of <i>C</i>; (ii) <i>A</i> and <i>B</i> are sub-skew left braces of <i>R</i>. 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 <i>n</i>-isoclinism is preserved from braces to groups. We also show that various nilpotency concepts on skew left braces are invariant under <i>n</i>-isoclinism.</p>

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Central series and (n)-isoclinism of skew left braces

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Arpan Kanrar,
  • Charlotte Roelants,
  • Manoj K. Yadav

摘要

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.