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A determinant for automorphisms of groups

Mattia Brescia

Abstract

Let $H$ and $K$ be groups. In this paper we introduce a concept of determinant for automorphisms of $H\times K$ and some concepts of incompatibility for group pairs as a measure of how much $H$ and $K$ are fare from being isomorphic. With the aid of the tools developed from these definitions, we give a characterisation of invertible automorphisms of $H\times K$ by means of their determinants and an explicit description of Aut($H\times K$) as a group of $2$-by-$2$ matrices, in case $H$ or $K$ belong to some relevant classes of groups. Many theoretical and practical applications of the determinants will be presented, together with examples and an analysis on some computational advantages of the determinants.

A determinant for automorphisms of groups

Abstract

Let and be groups. In this paper we introduce a concept of determinant for automorphisms of and some concepts of incompatibility for group pairs as a measure of how much and are fare from being isomorphic. With the aid of the tools developed from these definitions, we give a characterisation of invertible automorphisms of by means of their determinants and an explicit description of Aut() as a group of -by- matrices, in case or belong to some relevant classes of groups. Many theoretical and practical applications of the determinants will be presented, together with examples and an analysis on some computational advantages of the determinants.
Paper Structure (19 sections, 37 theorems, 61 equations)

This paper contains 19 sections, 37 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. A roadmap
  3. Preliminaries
  4. The determinants
  5. The general case: a sketch
  6. Group pairs
  7. Incompatible pairs
  8. Totally incompatible pairs
  9. Some Applications
  10. Groups satisfying Max-n and Min-n
  11. Abelian groups
  12. Stem groups
  13. Central automorphisms
  14. Computing inverse automorphisms
  15. Examples
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $H$ and $K$ be monoids. Then $\mathop{\rm End}(H\times K)\simeq \mathcal{M}_{H,K}$.

Theorems & Definitions (69)

  • Proposition 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • ...and 59 more