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IBIS primitive groups of almost simple type

Fabio Mastrogiacomo, Pablo Spiga

TL;DR

This work completes a focused classification of IBIS behavior among finite almost simple primitive groups with Lie-type socles, proving that IBIS occurs only in a small, explicit set of actions: $G\cong \mathrm{SL}_d(2)$ or $\mathrm{Sp}_d(2)$ acting on nonzero vectors over $\mathbb{F}_2$, or $G$ with socle $\Omega_d^{\pm}(q)$ acting on non-singular $1$-spaces with $d\ge 4$ and $q\ge 4$. The authors develop a robust framework based on standard vs non-standard actions, leveraging the Cameron–Kantor/Lie-type base-size bounds and a detailed case analysis across PSL, PSU, PSp, and orthogonal families, augmented by the Klein correspondence to translate between subspace actions. Their results rely on deep prior work (e.g., Liebeck–Shalev, Burness, LeSp, LuMoMo) to bound base sizes in non-standard actions and to identify the few exceptional IBIS instances in standard or low-rank cases. The paper thus narrows the IBIS landscape for almost-simple primitive groups of Lie type, providing a precise map of where irredundant bases share a uniform size and where multiple base sizes inevitably arise, with implications for base-size problems and related permutation-group structures. The findings have potential utility in computational group theory, as they delineate when irredundant bases behave like bases of a matroid, enabling efficient base-based representations and analyses in the identified cases.

Abstract

Let $G$ be a finite permutation group on $Ω$. An ordered sequence $(ω_1\ldots,ω_\ell)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. The minimal cardinality of a base is said to be the base size of $G$. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. In this paper, we classify the finite almost simple primitive IBIS groups whose base size is at least $6$.

IBIS primitive groups of almost simple type

TL;DR

This work completes a focused classification of IBIS behavior among finite almost simple primitive groups with Lie-type socles, proving that IBIS occurs only in a small, explicit set of actions: or acting on nonzero vectors over , or with socle acting on non-singular -spaces with and . The authors develop a robust framework based on standard vs non-standard actions, leveraging the Cameron–Kantor/Lie-type base-size bounds and a detailed case analysis across PSL, PSU, PSp, and orthogonal families, augmented by the Klein correspondence to translate between subspace actions. Their results rely on deep prior work (e.g., Liebeck–Shalev, Burness, LeSp, LuMoMo) to bound base sizes in non-standard actions and to identify the few exceptional IBIS instances in standard or low-rank cases. The paper thus narrows the IBIS landscape for almost-simple primitive groups of Lie type, providing a precise map of where irredundant bases share a uniform size and where multiple base sizes inevitably arise, with implications for base-size problems and related permutation-group structures. The findings have potential utility in computational group theory, as they delineate when irredundant bases behave like bases of a matroid, enabling efficient base-based representations and analyses in the identified cases.

Abstract

Let be a finite permutation group on . An ordered sequence of elements of is an irredundant base for if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. The minimal cardinality of a base is said to be the base size of . If all irredundant bases of have the same cardinality, is said to be an IBIS group. In this paper, we classify the finite almost simple primitive IBIS groups whose base size is at least .
Paper Structure (21 sections, 42 theorems, 186 equations, 1 table)

This paper contains 21 sections, 42 theorems, 186 equations, 1 table.

Table of Contents

  1. Introduction
  2. Standard and non-standard actions
  3. Structure of the paper
  4. Preliminaries
  5. Primitive action on totally-singular subspaces: case \ref{['HiHo1']} and case \ref{['HiHo3+']}
  6. Linear groups
  7. Case \ref{['pslk']}
  8. Case \ref{['psk1']}
  9. Case \ref{['psk2']}
  10. Unitary groups
  11. Symplectic groups
  12. The Klein correspondence
  13. Orthogonal groups of plus type
  14. Odd dimensional orthogonal groups
  15. Orthogonal groups of minus type
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $G$ be a primitive IBIS group. Then one of the following holds: Moreover, $G$ is a primitive IBIS group of diagonal type if and only if it belongs to the infinite family of diagonal groups $\{\mathrm{PSL}_2(2^f)\times \mathrm{PSL}_2(2^f)\mid f\in\mathbb{N},f\ge 2\}$ having degree $|\mathrm{PSL}_2(2^f)|=2^f(4^f-1)$.

Theorems & Definitions (75)

  • Theorem 1.1: Theorem 1.1, LuMoMo
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 65 more