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Burness, Timothy C.; Guralnick, Robert M.; Saxl, Jan

Base sizes for \(\mathcal S\)-actions of finite classical groups. (English) Zbl 1321.20002

Isr. J. Math. 199, Part B, 711-756 (2014).
Authors’ summary: Let \(G\) be a permutation group on a set \(\Omega\). A subset \(B\) of \(\Omega\) is a base for \(G\) if the pointwise stabilizer of \(B\) in \(G\) is trivial; the base size of \(G\) is the minimal cardinality of a base for \(G\), denoted by \(b(G)\). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher’s collection \(\mathcal S\) of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of \(\Omega\) form a base for \(G\). Indeed, with some specific exceptions, we show that almost all pairs of points in \(\Omega\) are bases.
Reviewer: Mohammad-Reza Darafsheh (Tehran)

Cited in 18 Documents

MSC:

20B15 Primitive groups
20G40 Linear algebraic groups over finite fields
20P05 Probabilistic methods in group theory

Keywords:

finite almost simple primitive permutation groups; point stabilizers; finite classical groups; base sizes; random subsets; probabilities

Software:

CTblLib; GAP Character Table Library; ATLAS Group Representations; GAP; Magma
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Full Text: DOI

References:

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