Groups without faithful transitive permutation representations of small degree
type: Item
instance of: scholarly article
title: Groups without faithful transitive permutation representations of small degree
zbMATH Open document ID: 0886.20020
author: László Babai
author: Albert J. Goodman
author: Laszló Pyber
published in: Journal of Algebra
publication date: 1998-03-08T00:00:00Z
review text: The authors investigate the titular groups, i.e., study the obstacles to a group \(G\) having large core-free subgroups \(H\). (\(H\) is core-free in \(G\) if it contains no normal subgroup \(\neq 1\) of \(G\).) Of particular interest are Dedekind subgroups of a group \(G\); a subgroup \(D\) of \(G\) is Dedekind if every subgroup of \(D\) is normal in \(G\). The main result of the paper is Theorem 4.9. There is a constant \(c\) such that the following holds with \(f(k)=\exp(c(\log k)^6)\): Each finite group \(G\) has two Dedekind subgroups \(D_1\) and \(D_2\) such that any subgroup of \(G\) of order exceeding \(f(k(G))\) has nontrivial intersection with \(D_1\) or \(D_2\), where \(k(G)\) denotes the order of the largest core-free subgroup of \(G\). Related properties of \(p\)-groups and infinite locally finite groups without large core-free subgroups are discussed, and as an extension of a result due to Gaschütz and Itô (a finite group \(G\) with \(k(G)=1\) is solvable) the authors obtain Theorem 2.13: There is a constant \(c\) such that if group \(G\) has a solvable subgroup of index \(n\) then \(G\) has a solvable normal subgroup of index at most \(n^c\). The authors end by summarizing ten open questions/conjectures mentioned in the body of the paper. This feature alone should attract many readers!
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1076264
zbMATH Keywords: large core-free subgroups
zbMATH Keywords: Dedekind subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: \(p\)-groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: solvable subgroups
reviewed by: W. E. Deskins
MaRDI profile type: Publication
full work available at URL: jabr.1997.7042
OpenAlex ID: W2088503029
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DOI: 10.1006/JABR.1997.7042
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description: scientific article; zbMATH DE number 1076264