Result: Class numbers of p-groups of a given order: Class numbers of \(p\)-groups of a given order.
Title:
Class numbers of p-groups of a given order: Class numbers of \(p\)-groups of a given order.
Authors:
Source:
Journal of Algebra. 279:810-819
Publisher Information:
Elsevier BV, 2004.
Publication Year:
2004
Subject Terms:
numbers of conjugacy classes, 0301 basic medicine, 03 medical and health sciences, finite \(p\)-groups, Algebra and Number Theory, Finite nilpotent groups, \(p\)-groups, 0101 mathematics, 01 natural sciences, Arithmetic and combinatorial problems involving abstract finite groups, Conjugacy classes for groups
Document Type:
Academic journal
Article
File Description:
application/xml
Language:
English
ISSN:
0021-8693
DOI:
10.1016/j.jalgebra.2004.03.006
Access URL:
Rights:
Elsevier Non-Commercial
Accession Number:
edsair.doi.dedup.....a82e70b8d2e404b9ee967fe61959d5f8
Database:
OpenAIRE
Further Information
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.