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Abstract
Let G be a permutation group, acting on a set Ω of size n. A subset B of Ω is a base for G if the pointwise stabilizer G_{(B) }is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r ≥ 1 such that Alt (m)^{r }... G ≤ Sym (m) \wr Sym (r), where the action of Sym (m) is on kelement subsets of {1,...,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M_{24} in its natural action on 24 points, or b(G) ≤ ⌈log n⌉ + 1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal.
Original language  English 

Number of pages  33 
Journal  Monatshefte für Mathematik 
Volume  First Online 
Early online date  5 Aug 2021 
DOIs  
Publication status  Epub ahead of print  5 Aug 2021 
Keywords
 Primitive groups
 Base size
 Classical groups
 Simple groups
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Dive into the research topics of 'Base sizes of primitive permutation groups'. Together they form a unique fingerprint.Projects
 1 Finished

CoDiMa: CoDiMa (CCP in the area of Computational Discrete Mathematics)
1/03/15 → 29/02/20
Project: Standard