Probabilistic Generation of Finite Almost Simple Groups
arxiv(2024)
摘要
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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