Central series' and ($n$)-isoclinism of skew left braces
Arpan Kanrar, Charlotte Roelants, Manoj K. Yadav
TL;DR
The article develops a higher-order isoclinism theory for skew left braces by introducing the class $\mathcal{I}_n$ and brace commutator words, enabling $n$-isoclinism within this framework. It proves that symmetric braces lie in $\mathcal{I}_n$ for all $n$ and establishes ambient embedding and Schur-like results that mirror classical group theory, including the construction of common ambient braces for isoclinic pairs and preservation of nilpotency properties under $n$-isoclinism. It connects brace theory with the cohomology of annihilator extensions via $H^2_b(K,A)$ and the transgression map, and shows that the natural semi-direct product $\Lambda_A$ preserves $n$-isoclinism, linking brace structure to group-theoretic invariants. The work also outlines substantial invariance results and poses open questions about the sizes of the $\mathcal{I}_n$-related classes, highlighting directions for further exploration in the interaction between brace theory and classical isoclinism concepts.
Abstract
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.