Class numbers of \(p\)-groups of a given order.
type: Item
instance of: scholarly article
title: Class numbers of \(p\)-groups of a given order.
zbMATH Open document ID: 1063.20020
author: Nigel Boston
author: I. M. Isaacs
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(p\) be a prime number, \(G\) a finite \(p\)-group and \(k(G)\) the class number of \(G\). Let \(D(p^e)\) denote the number of different numbers \(k(G)\) that arise as \(G\) runs over all groups of order \(p^e\), \(e\in\mathbb{N}\). In this paper, the authors study the behavior of the number \(D(p^e)\) as \(p\) varies while the exponent \(e\) is held constant. It is known that \(D(p^e)\) is bounded for \(e\leq 6\). According to E. O'Brien, \(D(p^7)\) is bounded; he also believes that \(D(p^8)\) is also bounded. Examples presented in the paper show that \(D(p^9)\) approaches infinity as \(p\) approaches infinity. In the proof some deep results of algebraic geometry are used.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 2104466
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: numbers of conjugacy classes
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.03.006
OpenAlex ID: W2001734675
cites work: Q4533411
cites work: Q3139838
cites work: Q3856819
cites work: Q4317891
cites work: Abelian varieties over finite fields
DOI: 10.1016/J.JALGEBRA.2004.03.006
description: scientific article; zbMATH DE number 2104466