On the normal subgroup with exactly two \(G\)-conjugacy class sizes.
type: Item
instance of: scholarly article
title: On the normal subgroup with exactly two \(G\)-conjugacy class sizes.
zbMATH Open document ID: 1213.20031
published in: Chinese Annals of Mathematics. Series B
publication date: 2009-12-15T00:00:00Z
review text: There have been many results over the last few years which relate the structure of groups to the sizes of conjugacy classes. The authors contribute to this study by proving the following interesting theorem: Let \(G\) be a finite group and \(R\) be a non-central normal subgroup of \(G\). Let \(N\) be a normal subgroup of \(G\) containing \(R\) such that each element of \(N\) has either \(1\) or \(m\) conjugates in \(G\), where \(m\) is some fixed natural number. Then \(N\) is nilpotent. By taking \(N\) to be the group \(G\), the old result of \textit{N. Itô} follows [Nagoya Math. J. 6, 17-28 (1953; Zbl 0053.01202)]. There are two comments which seem to the reviewer to be worth making. It is a straightforward result that in any transitive permutation group there is an element which acts without fixed points. In this paper there is no need to use the much stronger result that there is an element of prime-power order with this property, a result which depends on the classification of finite simple groups. The second comment is whether the condition that \(N\) contains a non-central Sylow subgroup is needed. The authors make no reference to this in the paper.
reviewed by: A. R. Camina
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 5647926
zbMATH Keywords: conjugacy class sizes
zbMATH Keywords: finite groups
zbMATH Keywords: numbers of conjugates
zbMATH Keywords: nilpotence
zbMATH Keywords: Sylow subgroups
author: Xianhe Zhao
author: Xiuyun Guo
MaRDI profile type: Publication
full work available at URL: s11401-008-0088-8
OpenAlex ID: W2067507634
cites work: Group Elements of Prime Power Index
cites work: Finite groups with two p-regular conjugacy class lengths
cites work: Implications of conjugacy class size
cites work: On the length of the conjugacy classes of finite groups
cites work: Q3905382
cites work: Character theory of finite groups
cites work: On Finite Groups with Given Conjugate Types I
cites work: SUBGROUPS WHICH ARE THE UNION OF FOUR CONJUGACY CLASSES
cites work: Subgroups which are the union of three conjugate classes
cites work: Q3824578
cites work: Q3500467
cites work: Finite groups with many conjugate elements
DOI: 10.1007/S11401-008-0088-8
Recommended article: A Note on Permutation Groups
Recommended article: Q3712476
Recommended article: On the number of fixed point free elements in a permutation group
Recommended article: Q3314037
Recommended article: Fixed-Point-Free p -Elements in Transitive Permutation Groups
Recommended article: Permutation 3-groups with no fixed-point-free elements.
Recommended article: Normal subgroups and class sizes of elements of prime power order.
Recommended article: On nilpotent subgroups containing non-trivial normal subgroups
Recommended article: Transitive permutation groups in which all derangements are involutions.
Recommended article: Permutation characters and fixed-point-free elements in permutation groups.
description: scientific article; zbMATH DE number 5647926