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Classifying finite groups G with three Aut(G)-orbits

Stephen P. Glasby

TL;DR

This work classifies finite groups $G$ for which $ ext{Aut}(G)$ acting on $G$ has exactly $3$ orbits. The authors reduce to a structural framework with $N=ig<G', ext{Φ}(G)ig>$, $V=G/N$, and $W=N$, where $A= ext{Aut}(G)^V rianglelefteq ext{GL}_m(r)$ and $B= ext{Aut}(G)ig|_W rianglelefteq ext{GL}_n(p)$ act transitively on the nonzero vectors of $V$ and $W$; Hering's classification of transitive linear groups drives the case analysis. They prove a three-case strategy based on the solvable radicals $A^ abla$ and $B^ abla$, and treat the $p$-group and $2$-group instances with explicit module and exterior-square constructions, yielding a complete, irredundant list of 3-orbit groups: one abelian, one non-nilpotent, three families of non-abelian $2$-groups, and two families of odd-$p$ non-abelian groups. The paper further explores 4-orbit groups, providing numerous examples and discussing the prospects for full classification in that regime, using exterior algebra and trace-based extensions to generate new families. Together, these results illuminate the solvable landscape of low-rank automorphism actions on finite groups and establish a framework for constructing and recognizing all 3-orbit groups—and potential pathways toward 4-orbit classifications.

Abstract

We give a complete and irredundant list of the finite groups $G$ for which Aut$(G)$, acting naturally on $G$, has precisely $3$ orbits. There are 7 infinite families: one abelian, one non-nilpotent, three families of non-abelian $2$-groups and two families of non-abelian $p$-groups with $p$ odd. The non-abelian $2$-group examples were first classified by Bors and Glasby in 2020 and non-abelian $p$-group examples with $p$ odd were classified independently by Li and Zhu, and by the author, in March 2024.

Classifying finite groups G with three Aut(G)-orbits

TL;DR

This work classifies finite groups for which acting on has exactly orbits. The authors reduce to a structural framework with , , and , where and act transitively on the nonzero vectors of and ; Hering's classification of transitive linear groups drives the case analysis. They prove a three-case strategy based on the solvable radicals and , and treat the -group and -group instances with explicit module and exterior-square constructions, yielding a complete, irredundant list of 3-orbit groups: one abelian, one non-nilpotent, three families of non-abelian -groups, and two families of odd- non-abelian groups. The paper further explores 4-orbit groups, providing numerous examples and discussing the prospects for full classification in that regime, using exterior algebra and trace-based extensions to generate new families. Together, these results illuminate the solvable landscape of low-rank automorphism actions on finite groups and establish a framework for constructing and recognizing all 3-orbit groups—and potential pathways toward 4-orbit classifications.

Abstract

We give a complete and irredundant list of the finite groups for which Aut, acting naturally on , has precisely orbits. There are 7 infinite families: one abelian, one non-nilpotent, three families of non-abelian -groups and two families of non-abelian -groups with odd. The non-abelian -group examples were first classified by Bors and Glasby in 2020 and non-abelian -group examples with odd were classified independently by Li and Zhu, and by the author, in March 2024.

Paper Structure

This paper contains 7 sections, 23 theorems, 23 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preparation for a proof of Theorem
  3. Case 1 of Theorem when
  4. Case 2 of Theorem when
  5. Case 3 of Theorem : and
  6. Examples of -orbit groups
  7. Examples of 4-orbit groups

Key Result

Theorem 1.1

Let $G$ be a finite $3$-orbit group with $N=\langle G',\Phi(G)\rangle$ and $|N|=p^n$. Then $1<N<G$ and $G$ is isomorphic to a group in lines $1-7$ of Table T:GVAWB. Moreover, the values of $V\cong G/N,A=\textup{Aut}(G)^V,W\cong N,B=\textup{Aut}(G)\downarrow W$ are valid, where $\textup{Aut}(G)^V$ an

Figures (1)

  • Figure 1: The $A$-submodules of $\Lambda^2{\mathcal{V}}$ where ${\mathcal{V}}=\mathbb{F}_{p^b}^{\,2\ell}$ and $\textup{Sp}({\mathcal{V}})\leqslant A\leqslant\Gamma\textup{Sp}({\mathcal{V}})$

Theorems & Definitions (56)

  • Theorem 1.1
  • Lemma 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 3.1
  • Definition 3.2
  • ...and 46 more