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Fixing two points in primitive solvable groups

Francesca Lisi, Luca Sabatini

TL;DR

This work analyzes two-point stabilizers in finite primitive solvable groups, proving that there exist two points whose stabilizer has derived length at most $9$, and that in the odd-order case the stabilizer can be chosen abelian (indeed cyclic in the quasiprimitive odd subcase). The authors build a framework linking two-point stabilizers to centralizers in the linear action $V times G$, and they leverage orbit structure results for linear groups, Gluck's permutation lemma, and a structured induction on $|V|+|G|$ to obtain a robust bound. The main contributions advance understanding of centralizer structure in solvable linear actions and refine the behavior of two-point stabilizers in primitive groups, with implications for orbit size questions and the interplay between Fitting/derived lengths. The results provide a principled approach to bounding stabilizer complexity and suggest directions for future improvements in constants and odd-order abelianness across broader classes of primitive groups.

Abstract

Consider a finite primitive solvable group. We observe that a result of Y. Yang implies that there exist two points whose pointwise stabilizer has derived length at most $9$. We show that, if the group has odd cardinality, then there exist two points whose pointwise stabilizer is abelian.

Fixing two points in primitive solvable groups

TL;DR

This work analyzes two-point stabilizers in finite primitive solvable groups, proving that there exist two points whose stabilizer has derived length at most , and that in the odd-order case the stabilizer can be chosen abelian (indeed cyclic in the quasiprimitive odd subcase). The authors build a framework linking two-point stabilizers to centralizers in the linear action , and they leverage orbit structure results for linear groups, Gluck's permutation lemma, and a structured induction on to obtain a robust bound. The main contributions advance understanding of centralizer structure in solvable linear actions and refine the behavior of two-point stabilizers in primitive groups, with implications for orbit size questions and the interplay between Fitting/derived lengths. The results provide a principled approach to bounding stabilizer complexity and suggest directions for future improvements in constants and odd-order abelianness across broader classes of primitive groups.

Abstract

Consider a finite primitive solvable group. We observe that a result of Y. Yang implies that there exist two points whose pointwise stabilizer has derived length at most . We show that, if the group has odd cardinality, then there exist two points whose pointwise stabilizer is abelian.
Paper Structure (5 sections, 9 theorems, 5 equations)

This paper contains 5 sections, 9 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Preliminaries
  4. The odd case
  5. The general case

Key Result

Theorem 1.1

Let $G$ be a primitive solvable group. Then there exist two points whose pointwise stabilizer has derived length at most $9$. If $|G|$ is odd, then there exist two points whose pointwise stabilizer is abelian.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2: Gluck's permutation lemma
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['th:mod']}(ii)
  • ...and 6 more