A note on the solvability of a finite group and the number of conjugate classes of its non-normal subgroups.
type: Item
instance of: scholarly article
title: A note on the solvability of a finite group and the number of conjugate classes of its non-normal subgroups.
zbMATH Open document ID: 1224.20048
author: Shunmin Chen
author: Guiyun Chen
published in: Southeast Asian Bulletin of Mathematics
publication date: 2011-09-29T00:00:00Z
review text: For a group \(G\), denote by \(\nu(G)\) the number of conjugacy classes of non-normal subgroups of \(G\). The following result is announced: Every finite group \(G\) with \(\nu(G)\leq 6\) is soluble. The proof of this contains a number of gaps. For example, Theorem 1.1 deals with a finite group \(G\) whose Sylow 2-subgroup \(S_2\) contains at most two \(G\)-conjugacy classes of subgroups that are non-normal in \(G\). The authors claim that either \(G\) is soluble or all proper subgroups of \(S_2\) are normal in \(G\). A counterexample for this is the alternating group \(A_5\). The proof of the aforementioned solubility result relies on Theorem 1.1. Reviewer's remark: The aforementioned result, however, is correct. It follows from the classification of all nonsoluble finite groups \(G\) with \(\nu(G)<14\); see [\textit{R. Brandl}, Beitr. Algebra Geom. 54, No. 2, 493--501 (2013; Zbl 1284.20022)].
reviewed by: Rolf Brandl
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5951294
zbMATH Keywords: numbers of non-normal subgroups
zbMATH Keywords: finite solvable groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: solvability
MaRDI profile type: Publication
Recommended article: A note on conjugacy class sizes and solvability of finite groups.
Recommended article: Q3640784
Recommended article: On The Number of Conjugacy Classes of a Finite Solvable Group
Recommended article: On Solvability of Finite Groups with Few Non-Normal Subgroups
Recommended article: On the number of conjugacy classes of a finite solvable group. II.
Recommended article: On the number of conjugacy classes of maximal subgroups in a finite soluble group
Recommended article: A note on groups with finite conjugacy classes of subnormal subgroups
Recommended article: Q3111460
Recommended article: The influence of the number of non-(sub)normal non-abelian subgroups on solvability of finite groups.
Recommended article: Bounds for the Number of Conjugacy Classes of Non-Normal Subgroups in a Finitep-Group
description: scientific article; zbMATH DE number 5951294