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Probabilistic Generation of Finite Almost Simple Groups

Jason Fulman, Daniele Garzoni, Robert M. Guralnick

Abstract

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension.

Probabilistic Generation of Finite Almost Simple Groups

Abstract

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension.
Paper Structure (17 sections, 28 theorems, 104 equations)

This paper contains 17 sections, 28 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Idea of the proof
  3. Notation
  4. Generating functions
  5. Cosets of $\operatorname{SL}_n(q)$ in $\operatorname{GL}_n(q)$
  6. Cosets of $\operatorname{SU}_n(q)$ in $\operatorname{GU}_n(q)$
  7. $\operatorname{Sp}_{2n}(q)$
  8. $\operatorname{O}^{\pm}_{2n}(q)$
  9. Group theory: preliminaries
  10. Expected number of fixed points
  11. Expectation for symmetric groups
  12. Classical groups: notation and limiting proportions
  13. Expectation for classical groups
  14. $\Omega_n(q)$ with $q$ odd
  15. A fixed point ratio estimate
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists an absolute constant $\epsilon>0$ such that the following holds. Let $S$ be a finite simple group of Lie type of large enough order, and let $x,y\in \operatorname{Aut}(S)$ with $x\neq 1$. Then the probability that $x$ and a random element of $Sy$ generate $\langle{S,x,y}\rangle$ is at l

Theorems & Definitions (54)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • ...and 44 more