The degree and index of a finite group
type: Item
instance of: scholarly article
title: The degree and index of a finite group
zbMATH Open document ID: 0932.20021
published in: Journal of Algebra
publication date: 1999-12-15T00:00:00Z
review text: For a finite group \(G\), denote by \(\delta(G)\) the least natural \(n\) such that \(G\) has a faithful permutation representation of degree \(n\) and by \(i(G)\) the minimal index of a proper subgroup of \(G\). The author proves that \(i(G)=\delta(G)\) iff \(G\) is simple (Theorem 1) and classifies all non-simple groups with \(2i(G)>\delta(G)\) (Theorem 6). A lot of similar results is obtained, for example, the following Proposition 11: Let \(p>2\) be the least prime divisor of \(G\). If \(\delta(G)>|G|/p^2\) then \(G\) has an ordered Sylow tower of the same type as supersolvable groups have.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20B35
zbMATH DE Number: 1295437
zbMATH Keywords: transitive degrees
zbMATH Keywords: normalized degrees
zbMATH Keywords: finite simple groups
zbMATH Keywords: faithful permutation representations
zbMATH Keywords: ordered Sylow towers
zbMATH Keywords: supersolvable groups
author: Yakov Berkovich
reviewed by: V. D. Mazurov
MaRDI profile type: Publication
full work available at URL: jabr.1998.7697
OpenAlex ID: W2031615993
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DOI: 10.1006/JABR.1998.7697
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description: scientific article; zbMATH DE number 1295437