Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The finite groups with three automorphism orbits

Cai Heng Li, Yan Zhou Zhu

TL;DR

This work completes the classification of finite 3-orbit groups by reducing to odd-p non-abelian p-groups and certain {p,q}-groups, then solving via a representation-theoretic framework on the pair (V,N/ Z(N)). The key methodology combines the exterior-square description of the center with transitive linear-group representations, yielding a dichotomy: either SL(3,q)-type or Sp(2m,q)-type actions govern the automorphism structure, leading to the families A_p(n,θ) (|θ|=3), extraspecial q-groups q^{1+2m}_{+} and their quotients, plus subfield-hyperplane quotients. The main result enumerates all 3-orbit groups, providing explicit automorphism-structure descriptions and connecting to rank-3 holomorphs, thereby solving Higman’s 1963 problem for finite groups and informing the finite rank-3 permutation group landscape. The findings offer concrete, algebraic constructions and criteria (e.g., transitivity on centers) to identify 3-orbit instances, with broader implications for fusions in finite group theory and combinatorial group actions.

Abstract

A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G. Higman in 1963. As a consequence, a classification is obtained for finite permutation groups of rank $3$ which are holomorphs of groups.

The finite groups with three automorphism orbits

TL;DR

This work completes the classification of finite 3-orbit groups by reducing to odd-p non-abelian p-groups and certain {p,q}-groups, then solving via a representation-theoretic framework on the pair (V,N/ Z(N)). The key methodology combines the exterior-square description of the center with transitive linear-group representations, yielding a dichotomy: either SL(3,q)-type or Sp(2m,q)-type actions govern the automorphism structure, leading to the families A_p(n,θ) (|θ|=3), extraspecial q-groups q^{1+2m}_{+} and their quotients, plus subfield-hyperplane quotients. The main result enumerates all 3-orbit groups, providing explicit automorphism-structure descriptions and connecting to rank-3 holomorphs, thereby solving Higman’s 1963 problem for finite groups and informing the finite rank-3 permutation group landscape. The findings offer concrete, algebraic constructions and criteria (e.g., transitivity on centers) to identify 3-orbit instances, with broader implications for fusions in finite group theory and combinatorial group actions.

Abstract

A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G. Higman in 1963. As a consequence, a classification is obtained for finite permutation groups of rank which are holomorphs of groups.
Paper Structure (10 sections, 29 theorems, 52 equations, 1 table)

This paper contains 10 sections, 29 theorems, 52 equations, 1 table.

Table of Contents

  1. Introduction
  2. $3$-orbit groups and special $p$-groups
  3. Some known finite $3$-orbit groups
  4. Special $p$-groups of exponent $p$
  5. Extraspecial $q$-groups
  6. Exterior Square of Transitive Linear Groups
  7. $3$-orbit $p$-groups for odd prime $p$
  8. ${\mathrm{SL}}(3,q)\lhd G$
  9. ${\mathrm{Sp}}(2m,p^n)\lhd G$
  10. Subfield hyperplanes and quotients of $q^{1+2m}_+$

Key Result

Theorem A

Let $V={\mathbb{F}}_p^n$ for odd prime $p$, $G\leqslant {\mathrm{GL}}(V)$ be non-solvable, and let $W<\Lambda^2_{{\mathbb{F}}_p}(V)$. If $G$ acts transitively on non-zero vectors of both $V$ and $\Lambda^2_{{\mathbb{F}}_p}(V)/W$, then one of the following statements holds.

Theorems & Definitions (53)

  • Theorem A
  • Theorem B
  • Corollary 1.1
  • Theorem C
  • Corollary 1.2
  • Example 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Example 2.4
  • Proposition 2.5
  • ...and 43 more