Orbit invariants and an application to the Baby Monster.
type: Item
instance of: scholarly article
title: Orbit invariants and an application to the Baby Monster.
zbMATH Open document ID: 1245.20015
author: Max Neunhöffer
author: Felix Noeske
author: E. A. O'Brien
author: Robert A. Wilson
published in: Journal of Algebra
publication date: 2012-02-11T00:00:00Z
review text: Let \(G\) be a permutation group acting on a set \(X\). A subset \(B\) of \(X\) is called a base for \(G\) if the pointwise stabilizer of \(B\) in \(G\) is trivial. Let \(b(G)\) be the minimal size of a base for a permutation group \(G\). Cameron and Kantor conjectured that there is a constant \(c\) such that \(b(G)\leq c\) for every almost simple group \(G\) in a faithful primitive non-standard action. This conjecture was proved by \textit{M. W. Liebeck} and \textit{A. Shalev}, [J. Am. Math. Soc. 12, No. 2, 497-520 (1999; Zbl 0916.20003)]. Subsequent research showed that \(b(G)\leq 7\) for every finite almost simple group \(G\). In the paper [Isr. J. Math. 177, 307-333 (2010; Zbl 1216.20008)], \textit{T. C. Burness, E. A. O'Brien} and \textit{R. A. Wilson} obtained the precise value of \(b(G)\) for the primitive action of all almost simple sporadic groups \(G\) missing two cases concerning the Baby Monster. Now in the paper under review the authors consider the missing two cases and in addition to the previous result they obtain the precise result as follows: Let \(B\) denote the Baby Monster group and consider it as acting on the right cosets of a maximal subgroup \(H\). If \(H=2^{2+10+20}.(M_{22}:2\times S_3)\), then \(b(B)=3\), and if \(H=2^{30}.L_5(2)\), then \(b(B)=2\).
reviewed by: M. R. Darafsheh
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20B40
zbMATH DE Number: 6005438
zbMATH Keywords: Baby Monster
zbMATH Keywords: orbit invariants
zbMATH Keywords: non-standard actions
zbMATH Keywords: base sizes
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: sporadic simple groups
zbMATH Keywords: bases of minimal size
describes a project that uses: GAP
describes a project that uses: ATLAS Group Representations
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.05.033
OpenAlex ID: W2004175737
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DOI: 10.1016/J.JALGEBRA.2011.05.033
description: scientific article; zbMATH DE number 6005438