Probabilistic Generation of Finite Almost Simple Groups

2024/03/26 by Jason Fulman, Fulman, Jason, Daniele Garzoni +3 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2403.17291

openalex publication_date 2024/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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• Derangements in finite classical groups and characteristic polynomials of random matrices

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