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Zero-one generation laws for finite simple groups. (English) Zbl 1515.20323

Summary: Let \(G\) be a simple algebraic group over the algebraic closure of \(\mathbb{F}_p\) (\(p\) prime), and let \(G(q)\) denote a corresponding finite group of Lie-type over \(\mathbb{F}_q\), where \(q\) is a power of \(p\). Let \(X\) be an irreducible subvariety of \(G^r\) for some \(r\geq 2\). We prove a zero-one law for the probability that \(G(q)\) is generated by a random \(r\)-tuple in \(X(q) = X\cap G(q)^r\): the limit of this probability as \(q\) increases (through values of \(q\) for which \(X\) is stable under the Frobenius morphism defining \(G(q)\)) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs \(G(q)\) to be generated by an \(r\)-tuple in \(X(q)\) for two sufficiently large values of \(q\). We also prove a version of this result where the underlying characteristic is allowed to vary.
In our main application, we apply these results to the case where \(r=2\) and the irreducible subvariety \(X = C\times D\), a product of two conjugacy classes of elements of finite order in \(G\). This leads to new results on random \((2,3)\)-generation of finite simple groups \(G(q)\) of exceptional Lie-type: provided \(G(q)\) is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate \(G(q)\) tends to \(1\) as \(q \rightarrow \infty \). Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and \(\mathrm{PSp}_4(q)\)) are randomly \((2,3)\)-generated.
Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie-type.

MSC:

20P05 Probabilistic methods in group theory
20G40 Linear algebraic groups over finite fields
20D06 Simple groups: alternating groups and groups of Lie type

Software:

CHEVIE
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Full Text: DOI arXiv

References:

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