Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable Extensions of Complete Groups

Isaac Ochoa

TL;DR

The paper develops a framework for constructing and classifying finite stable groups that are nontrivial extensions of centerless groups. By leveraging complete groups and the structure of extensions with trivial outer action, it proves splitting results that force central extensions to decompose as direct products, and it derives precise criteria for stability in these settings. It provides a finite-classification result: the only finite stable central extensions of centerless groups have kernel of order $2$, with stability characterized by a complete quotient $K$ whose abelianization has a nontrivial cyclic Sylow $2$-subgroup; in the nilpotency-class-two setting the only finite stable group is $D_4$, yielding stability for $D_4\times K$ under suitable conditions on $K$. The paper then constructs infinite families of complete groups via affine semilinear groups $K_q$, producing infinitely many stable examples such as $C_2\times K_q$ and $D_4\times K_q$, and demonstrates corollaries including infinitely many non-stable finite groups equinumerous with their automorphism groups. Overall, the work offers explicit, scalable methods to generate and classify stable yet non-complete groups and highlights rich interactions between kernel structure, quotient completeness, and outer actions.

Abstract

A group is said to be stable if it is isomorphic to its automorphism group. Centerless groups are naturally embedded in their automorphism groups via the map sending an element to conjugation by that element, partially constraining the structure of their automorphisms. As such, it is natural to ask if we can use centerless groups to construct stable groups with nontrivial centers. To this end, we classify all finite stable groups arising as central extensions of centerless groups. Furthermore, all finite stable groups arising as extensions of centerless groups by groups of nilpotency class two with trivial induced outer action on the kernel are classified. Finally, it is shown that there are infinitely many stable groups of each of the above two types. As a corollary we show that there are infinitely many non-stable finite groups equinumerous with their automorphism groups.

Stable Extensions of Complete Groups

TL;DR

The paper develops a framework for constructing and classifying finite stable groups that are nontrivial extensions of centerless groups. By leveraging complete groups and the structure of extensions with trivial outer action, it proves splitting results that force central extensions to decompose as direct products, and it derives precise criteria for stability in these settings. It provides a finite-classification result: the only finite stable central extensions of centerless groups have kernel of order , with stability characterized by a complete quotient whose abelianization has a nontrivial cyclic Sylow -subgroup; in the nilpotency-class-two setting the only finite stable group is , yielding stability for under suitable conditions on . The paper then constructs infinite families of complete groups via affine semilinear groups , producing infinitely many stable examples such as and , and demonstrates corollaries including infinitely many non-stable finite groups equinumerous with their automorphism groups. Overall, the work offers explicit, scalable methods to generate and classify stable yet non-complete groups and highlights rich interactions between kernel structure, quotient completeness, and outer actions.

Abstract

A group is said to be stable if it is isomorphic to its automorphism group. Centerless groups are naturally embedded in their automorphism groups via the map sending an element to conjugation by that element, partially constraining the structure of their automorphisms. As such, it is natural to ask if we can use centerless groups to construct stable groups with nontrivial centers. To this end, we classify all finite stable groups arising as central extensions of centerless groups. Furthermore, all finite stable groups arising as extensions of centerless groups by groups of nilpotency class two with trivial induced outer action on the kernel are classified. Finally, it is shown that there are infinitely many stable groups of each of the above two types. As a corollary we show that there are infinitely many non-stable finite groups equinumerous with their automorphism groups.
Paper Structure (5 sections, 19 theorems, 137 equations)

This paper contains 5 sections, 19 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Notation & Background
  3. Stable Central Extensions of Complete Groups
  4. Extensions by Groups of Nilpotency Class Two
  5. Affine Semilinear Groups of Degree One

Key Result

Lemma 3.1

Let $1\rightarrow N\rightarrow G\xrightarrow{\pi} K\rightarrow 1$ be an extension such that the induced outer action on $N$ is trivial. Identifying $N$ with its image in $G$, if $N$ is nilpotent and $K$ is centerless, then:

Theorems & Definitions (39)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 29 more