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The category of formations of finite groups and topology

Ismael Gutierrez Garcia, Luz Adriana Mejía Castaño

Abstract

This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete posetal category of group classes, $\mathrm{CG}$. Second, we formalize the collection of closure operators themselves as a category, \textbf{CL}, proving it is a complete lattice. This provides the essential machinery for combining algebraic operations and understanding their universal properties via adjunctions. Finally, we apply this framework to topology. We show that additive universally anchored operators induce homotopically equivalent contractible spaces, revealing a principle of global simplicity that contrasts with local algebraic friction. We then use the lattice structure of \textbf{CL} to analyze the operators for Formations and Fitting classes, uncovering a profound topological asymmetry between these dually defined structures.

The category of formations of finite groups and topology

Abstract

This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete posetal category of group classes, . Second, we formalize the collection of closure operators themselves as a category, \textbf{CL}, proving it is a complete lattice. This provides the essential machinery for combining algebraic operations and understanding their universal properties via adjunctions. Finally, we apply this framework to topology. We show that additive universally anchored operators induce homotopically equivalent contractible spaces, revealing a principle of global simplicity that contrasts with local algebraic friction. We then use the lattice structure of \textbf{CL} to analyze the operators for Formations and Fitting classes, uncovering a profound topological asymmetry between these dually defined structures.
Paper Structure (9 sections, 7 theorems, 6 equations, 1 figure)

This paper contains 9 sections, 7 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The category of group classes and the monoidal category of formations
  4. A Categorical perspective on closure operators
  5. Global Topological Invariance versus Local Algebraic Difference
  6. Algebraic Interpretation of Topological Invariance
  7. A Tale of Two Assymetries: The Structural Regularization in Formations
  8. Conclusion
  9. Funding declaration

Key Result

Lemma 3.5

Let $\mathfrak{X}$ be an $\hbox{\scriptsize\sf R}_0$-closed class and $G$ a finite group. Then the set partially ordered by inclusion, has a unique minimal element, denoted by $G^{\mathfrak{X}}$ and called the $\mathfrak{X}$-residual of $G$. It is a characteristic subgroup of $G$, and if $\mathfrak{X}$ is a formation and $\varepsilon: G \twoheadrightarrow \varepsilon(G)$ is an epimorphism, then $

Figures (1)

  • Figure 1: Graph of some morphisms in the category $\mathrm{CL}$

Theorems & Definitions (25)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Definition 3.6
  • Proposition 3.7
  • proof
  • Theorem 3.8
  • ...and 15 more