On groups with no regular orbits on the set of subsets
type: Item
instance of: scholarly article
title: On groups with no regular orbits on the set of subsets
zbMATH Open document ID: 0575.20002
DOI: 10.1007/BF01196649
author: Peter J. Cameron
author: Peter M. Neumann
author: Jan Saxl
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: Let G be a permutation group on a finite set \(\Omega\) of size n. Then G acts naturally on the set P(\(\Omega)\) of all subsets of \(\Omega\). In the present paper the following theorem is proved. Let G be primitive on \(\Omega\) of degree n, G neither alternating nor symmetric. Then, with only finitely many exceptions, G has a regular orbit on P(\(\Omega)\). The proof is based on the fact that if G is transitive of degree n then bm\(\geq n\), where m is the minimal degree of G on \(\Omega\) and b is the base size of G on \(\Omega\), and on order bounds for G obtained using the classification of finite simple groups [cf. \textit{P. J. Cameron}, Bull. Lond. Math. Soc. 13, 1-22 (1981; Zbl 0463.20003)]. Without the classification the theorem can be proved for 2-transitive G using \textit{L. Babai}'s bounds [Invent. Math. 65, 473-484 (1982; Zbl 0478.20002)]. It remains an open question whether a more elementary proof of the theorem can be given for simply primitive groups. The theorem implies some interesting consequences on some recent work of \textit{N. Inglis} [Arch. Math. 43, 297-300 (1984; Zbl 0545.20001)] and Siemons and Wagner (to appear).
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20B20
zbMATH DE Number: 3918627
zbMATH Keywords: finite primitive permutation groups
zbMATH Keywords: action on subsets
zbMATH Keywords: regular orbit
zbMATH Keywords: minimal degree
zbMATH Keywords: order bounds
zbMATH Keywords: classification of finite simple groups
zbMATH Keywords: simply primitive groups
MaRDI profile type: Publication
cites work: On the order of uniprimitive permutation groups
cites work: On the order of doubly transitive permutation groups
cites work: Finite Permutation Groups and Finite Simple Groups
cites work: On orbit equivalent permutation groups
cites work: On finite permutation groups with the same orbits on unordered sets
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description: scientific article; zbMATH DE number 3918627