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On $n$-isoclinism of skew braces

Risa Arai, Cindy Tsang

TL;DR

This work extends the group-theoretic notion of n-isoclinism to skew braces by developing a verbal/marginal framework tailored to the dual-operation structure of braces. It introduces verbal left/right series and verbal sub-skew braces, along with marginal left ideals, to provide robust invariants for isoclinism that avoid issues with naive star-product-based definitions. A new W-isoclinism is proposed, using Core$_A$(M_W(A)) to form quotients and commuting diagrams for skew-brace words, with the strength of the notion decreasing as n grows. The paper also provides concrete brace examples showing genuine differences between quotient constructions and connects the new framework to classical isoclinism in suitable special cases.

Abstract

The purpose of this paper is to explore possible definitions of $n$-isoclinism for skew braces. We also introduce the notions of verbal sub-skew braces and marginal left ideals.

On $n$-isoclinism of skew braces

TL;DR

This work extends the group-theoretic notion of n-isoclinism to skew braces by developing a verbal/marginal framework tailored to the dual-operation structure of braces. It introduces verbal left/right series and verbal sub-skew braces, along with marginal left ideals, to provide robust invariants for isoclinism that avoid issues with naive star-product-based definitions. A new W-isoclinism is proposed, using Core(M_W(A)) to form quotients and commuting diagrams for skew-brace words, with the strength of the notion decreasing as n grows. The paper also provides concrete brace examples showing genuine differences between quotient constructions and connects the new framework to classical isoclinism in suitable special cases.

Abstract

The purpose of this paper is to explore possible definitions of -isoclinism for skew braces. We also introduce the notions of verbal sub-skew braces and marginal left ideals.

Paper Structure

This paper contains 9 sections, 13 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Verbal subgroups and marginal subgroups
  3. Verbal subgroups
  4. Marginal subgroups
  5. Definition of $n$-isoclinism
  6. Verbal sub-skew braces and marginal left ideals
  7. Verbal sub-skew braces
  8. Marginal left ideals
  9. Definition of $n$-isoclinism for skew braces

Key Result

Proposition 2.2

Let $G$ be a group. For any $n\geq 1$, we have

Theorems & Definitions (38)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 28 more