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Isbell's subfactor projections in a noetherian form

Kishan Dayaram, Amartya Goswami, Zurab Janelidze

TL;DR

Isbell's subfactor theory and the associated projection machinery are reexamined within the noetherian-form framework, enabling a general, functorial treatment of the butterfly lemma and the refinement theorem. The authors identify a concrete flaw in Isbell's coarsest-refinement claim but show that the remaining results extend to self-dual noetherian forms, yielding canonical isomorphisms via Homomorphism Induction and Universal Isomorphism theorems. They establish the Subfactor Projection Lemma and derive the butterfly lemma in this abstract setting, with applicability to semi-abelian and Grandis exact categories, while also providing a counterexample to the coarsest-refinement claims. The work situates these results within a broader landscape of abstract approaches, clarifying the scope and limitations of generalizations beyond classical group theory.

Abstract

In this paper, we revisit the 1979 work of Isbell on subfactors of groups and their projections, which he uses to establish a stronger formulation of the butterfly lemma and its consequence, the refinement theorem for subnormal series of subgroups. We point out an error in the second part of Isbell's refinement theorem, but show that the rest of his results can be extended to the general self-dual context of a noetherian form, which includes in its scope all semi-abelian categories as well as all Grandis exact categories. Furthermore, we show that Isbell's formulations of the butterfly lemma and the refinement theorem amount to canonicity of isomorphisms established in these results.

Isbell's subfactor projections in a noetherian form

TL;DR

Isbell's subfactor theory and the associated projection machinery are reexamined within the noetherian-form framework, enabling a general, functorial treatment of the butterfly lemma and the refinement theorem. The authors identify a concrete flaw in Isbell's coarsest-refinement claim but show that the remaining results extend to self-dual noetherian forms, yielding canonical isomorphisms via Homomorphism Induction and Universal Isomorphism theorems. They establish the Subfactor Projection Lemma and derive the butterfly lemma in this abstract setting, with applicability to semi-abelian and Grandis exact categories, while also providing a counterexample to the coarsest-refinement claims. The work situates these results within a broader landscape of abstract approaches, clarifying the scope and limitations of generalizations beyond classical group theory.

Abstract

In this paper, we revisit the 1979 work of Isbell on subfactors of groups and their projections, which he uses to establish a stronger formulation of the butterfly lemma and its consequence, the refinement theorem for subnormal series of subgroups. We point out an error in the second part of Isbell's refinement theorem, but show that the rest of his results can be extended to the general self-dual context of a noetherian form, which includes in its scope all semi-abelian categories as well as all Grandis exact categories. Furthermore, we show that Isbell's formulations of the butterfly lemma and the refinement theorem amount to canonicity of isomorphisms established in these results.
Paper Structure (5 sections, 14 theorems, 32 equations, 1 table)

This paper contains 5 sections, 14 theorems, 32 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isbell's results on subfactors
  4. Counterexample to Isbell's claim on coarsest refinements
  5. On other abstract approaches to the butterfly lemma

Key Result

Lemma 1.1

The projection of a subfactor $(Y,Y^+)$ in a subfactor $(X,X^+)$ is a subfactor $(M,M^+)$, and $(M,M^+)$ and $(X,X^+)$ have the same projection in $(Y,Y^+).$

Theorems & Definitions (21)

  • Lemma 1.1
  • Theorem 2.3: Lattice Isomorphism Theorem
  • Theorem 2.4: Restricted Modular Law, dnaiv
  • Theorem 2.7: Homomorphism Induction Theorem, dnaiv
  • Corollary 2.8: Universal Isomorphism Theorem
  • Lemma 2.9
  • proof
  • Lemma 2.10: dnaiv
  • Lemma 2.11: dnaiv
  • Corollary 2.12: Less Restricted Modular Law
  • ...and 11 more