Year: 2021
Type: article
Abstract: LetGbe a permutation group, acting on a set $$\varOmega $$ Ω of sizen. A subset $${\mathcal {B}}$$ B of $$\varOmega $$ Ω is abaseforGif the pointwise stabilizer $$G_{({\mathcal {B}})}$$ G(B) is trivial. Letb(G) be the minimal size of a base forG. A subgroupGof $$\mathrm {Sym}(n)$$ Sym(n) islarge baseif there exist integersmand $$r \ge 1$$ r≥1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ Alt(m)r⊴G≤Sym(m)≀Sym(r) , where the action of $${{\,\mathrm{Sym}\,}}(m)$$ Sym(m) is onk-element subsets of $$\{1,\dots ,m\}$$ {1,⋯,m} and the wreath product acts with product action. In this paper we prove that ifGis primitive and not large base, then eitherGis the Mathieu group $$\mathrm {M}_{24}$$ M24 in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ b(G)≤⌈logn⌉+1 . Furthermore, we show that there are infinitely many primitive groupsGthat are not large base for which $$b(G) > \log n + 1$$ b(G)>logn+1 , so our bound is optimal.
Source: Monatshefte für Mathematik
Institutions: University of Bologna, University of St Andrews
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,780
OpenAlex ID: W3172749101