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arXiv · 2403.17291

Probabilistic Generation of Finite Almost Simple Groups

Jason Fulman·Daniele Garzoni·Robert M. Guralnick
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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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Jason Fulman, Daniele Garzoni, Robert M. Guralnick. 2024-03-26. Probabilistic Generation of Finite Almost Simple Groups. https://arxiv.org/abs/2403.17291

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