The structure of finite \(p\)-groups: Effective proof of the coclass conjectures
type: Item
instance of: scholarly article
title: The structure of finite \(p\)-groups: Effective proof of the coclass conjectures
zbMATH Open document ID: 0795.20009
DOI: 10.1007/BF01231763
published in: Inventiones Mathematicae
publication date: 1994-09-08T00:00:00Z
full work available at URL: https://eudml.org/doc/144172
review text: In 1980 C. R. Leedham-Green and M. F. Newman proposed a program for classifying finite \(p\)-groups using the coclass as the primary invariant. The coclass of a group of order \(p^ n\) and nilpotency class \(c\) is defined to be \(n-c\); a pro-\(p\) group of coclass \(r\) is an inverse limit of finite \(p\)-groups of coclass \(r\). Thus, groups of coclass 1 are exactly the groups of maximal class studied by N. Blackburn and many others. Let us now state the coclass conjectures, ordered by increasing strength (that is, each conjecture implies the previous ones). Conjecture E. Given \(p\) and \(r\), there are only finitely many isomorphism types of infinite soluble pro-\(p\) groups of coclass \(r\). Conjecture D. Given \(p\) and \(r\), there are only finitely many isomorphism types of infinite pro-\(p\) groups of coclass \(r\). Conjecture C. Pro-\(p\) groups of finite coclass are soluble. Conjecture B. For some function \(g\), every \(p\)-group of coclass \(r\) has derived length at most \(g(p,r)\). Conjecture A. For some function \(f\), every \(p\)-group of coclass \(r\) has a normal subgroup \(N\) of class at most 2 (1, if \(p=2\)) and index at most \(f(p,r)\). Some of these conjectures may be reformulated as combinatorial statements on the structure of certain graphs. Thus, let \(T_{p,r}\) be a (directed) tree whose vertices are all \(p\)-groups of coclass \(r\), where \((H,G)\) is an edge if there is an epimorphism from \(H\) to \(G\) whose kernel has order \(p\). Then Conjecture D is tantamount to saying that there are only finitely many infinite chains in \(T_{p,r}\). Over the past 15 years the coclass conjectures attracted the attention of various mathematicians, such as Leedham-Green, S. Donkin, S. McKay, W. Plesken, A. Mann, E. I. Zelmanov, M. F. Newman, E. A. O'Brien, and others. The combination of Donkin's and Leedham-Green's important contributions settles Conjecture A in the affirmative for primes \(p>3\). This approach, which is based on algebraic groups…
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 549848
zbMATH Keywords: classification of finite \(p\)-groups
zbMATH Keywords: nilpotency class
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: inverse limits of finite \(p\)-groups
zbMATH Keywords: groups of maximal class
zbMATH Keywords: coclass conjectures
zbMATH Keywords: infinite soluble pro-\(p\) groups
zbMATH Keywords: derived lengths
zbMATH Keywords: powerful \(p\)-groups
zbMATH Keywords: uniserial actions
zbMATH Keywords: modular Lie algebras
Wikidata QID: Q59541317
author: Aner Shalev
MaRDI profile type: Publication
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OpenAlex ID: W2085522287
description: scientific article; zbMATH DE number 549848