On \(p\)-groups having the minimal number of conjugacy classes of maximal size.
type: Item
instance of: scholarly article
title: On \(p\)-groups having the minimal number of conjugacy classes of maximal size.
zbMATH Open document ID: 1190.20010
author: M. F. Newman
author: E. A. O'Brien
author: Andrei Jaikin-Zapirain
published in: Israel Journal of Mathematics
publication date: 2009-10-02T00:00:00Z
review text: There is a problem of I. D. Macdonald of existence of a finite \(p\)-group of odd order with exactly \(p-1\) conjugacy classes of maximal size (e.g., Problem 14.92 in The Kourovka Notebook). The authors produce a 3-group with this property. In fact, they prove two general results (which combine only for \(p=3\)). A conjugacy class of a finite \(p\)-group is called special if it is equal to a coset of the Frattini subgroup; a special class is always of maximal size, and if there is a special class, then all classes of maximal size are special. Theorem~1: For any odd prime \(p\) there is a group of order \(p^{2p+2}\) with exactly \((p-1)^2\) special conjugacy classes. Theorem~2: If there is a finite non-cyclic \(p\)-group with exactly \(2(p-1)\) special conjugacy classes, then there is a finite \(p\)-group with exactly \(p-1\) special conjugacy classes.
reviewed by: Evgenii I. Khukhro
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 5610600
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: numbers of conjugacy classes
zbMATH Keywords: conjugacy classes of maximal size
zbMATH Keywords: Frattini subgroup
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s11856-009-0066-1
OpenAlex ID: W1973169717
cites work: The Magma algebra system. I: The user language
cites work: Endliche Gruppen I
cites work: Finite P-groups with unique maximal classes
cites work: Groups with few class sizes and the centraliser equality subgroup.
cites work: Q4285784
DOI: 10.1007/S11856-009-0066-1
description: scientific article; zbMATH DE number 5610600