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Quasirandom and quasisimple groups

Marco Barbieri, Luca Sabatini

TL;DR

The paper develops a structural theory of finite ε-quasirandom groups, showing that, for fixed ε>0 and sufficiently large size, G/Sol(G) must be a bounded direct product of finite simple groups of Lie type with bounded rank. It then provides a complete classification of 1/5-quasirandom groups, showing that such groups are either of two affine types or quasisimple, with sharp corollaries like every 3/14-quasirandom group being quasisimple. The method combines quantitative representation-theoretic bounds with a detailed analysis of linear and projective representations, the Schur multiplier, and known data for simple groups, supplemented by an affine-type construction yielding highly quasirandom nonquasisimple examples. The results have implications for understanding expansion phenomena and the rigidity of quasirandom groups, clarifying when highly quasirandom behavior forces strong simple- or almost-simple structure. The paper also provides concrete, computable classifications in the exponential quasirandom regime and contributes explicit examples near the threshold values that delineate quasisimplicity from affine-type behavior.

Abstract

Fix $\varepsilon > 0$. We say that a finite group $G$ is $\varepsilon$-quasirandom if every nontrivial irreducible complex representation of $G$ has degree at least $|G|^\varepsilon$. In this paper, we give a structure theorem for large $\varepsilon$-quasirandom groups, and we completely classify the $\frac{1}{5}$-quasirandom groups.

Quasirandom and quasisimple groups

TL;DR

The paper develops a structural theory of finite ε-quasirandom groups, showing that, for fixed ε>0 and sufficiently large size, G/Sol(G) must be a bounded direct product of finite simple groups of Lie type with bounded rank. It then provides a complete classification of 1/5-quasirandom groups, showing that such groups are either of two affine types or quasisimple, with sharp corollaries like every 3/14-quasirandom group being quasisimple. The method combines quantitative representation-theoretic bounds with a detailed analysis of linear and projective representations, the Schur multiplier, and known data for simple groups, supplemented by an affine-type construction yielding highly quasirandom nonquasisimple examples. The results have implications for understanding expansion phenomena and the rigidity of quasirandom groups, clarifying when highly quasirandom behavior forces strong simple- or almost-simple structure. The paper also provides concrete, computable classifications in the exponential quasirandom regime and contributes explicit examples near the threshold values that delineate quasisimplicity from affine-type behavior.

Abstract

Fix . We say that a finite group is -quasirandom if every nontrivial irreducible complex representation of has degree at least . In this paper, we give a structure theorem for large -quasirandom groups, and we completely classify the -quasirandom groups.
Paper Structure (11 sections, 20 theorems, 73 equations, 5 tables)

This paper contains 11 sections, 20 theorems, 73 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal degree of a nontrivial representation
  4. Quasirandom degree
  5. Quasisimple groups
  6. Linear and projective representations
  7. Quasirandom groups of affine type
  8. Proof of the main theorems
  9. Proof of Theorem \ref{['thm:A']}
  10. Proof of Theorem \ref{['thm:B']}
  11. Appendix: Quasirandom degree of finite simple groups

Key Result

Theorem 1

Fix $\varepsilon>0$. If $G$ is an $\varepsilon$-quasirandom group, then $|G/\mathrm{Sol}(G)| \geq |G|^\varepsilon$. If $|G|$ is sufficiently large depending on $\varepsilon$, then $G/\mathrm{Sol}(G)$ is a direct product of boundedly many finite simple groups of Lie type of bounded rank. Conversely,

Theorems & Definitions (46)

  • Definition
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • ...and 36 more