Infinite generation of automorphism groups of free pro-\(p\) groups
type: Item
instance of: scholarly article
title: Infinite generation of automorphism groups of free pro-\(p\) groups
zbMATH Open document ID: 0824.20031
DOI: 10.1007/BF00975176
published in: Siberian Mathematical Journal
publication date: 1994-11-08T00:00:00Z
review text: The author considers the automorphism group of a free pro-\(p\) group as well as the automorphism group of a free profinite group \(F_ n\). The results concern uniformly saturated pro-\(p\) groups. It is proved that if \(G_ 0\) is a saturated pro-\(p\) group, then the completed group algebra \(\widehat {S} =Z/_ p ZG_ 0\) satisfies the maximal condition for closed right ideals. The main theorem states that the automorphism group of the free metabelian pro-\(p\) group \(M_ n (p)\) of rank \(n \geq 2\) is infinitely generated (i.e. there is no finite generating set, in the topological sense). From that it follows as a corollary, that the automorphism group of the free pro-\(p\) group \(F_ n (p)\) of rank \(n \geq 2\) is infinitely generated and the same for the automorphism group \(\text{Aut }F_ n\) of the free profinite group \(F_ n\) of finite rank \(n \geq 2\). Another application of the results of the author is that if \(N_{n,m}\) (resp. \(M_{n,m}\)) is the free nilpotent group (free metabelian nilpotent group) of rank \(n\) and class \(m\), then the number of generators for the groups \(\text{Aut }N_{n,m}\) and \(\text{Aut }M_{n,m}\) increases without bound as \(m\) increases for \(n \geq 2\). The proofs are described via \(p\)-adic analytic groups. Details of the terminology and symbolism are difficult to be given here.
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 652183
zbMATH Keywords: automorphism groups
zbMATH Keywords: free pro-\(p\) groups
zbMATH Keywords: free profinite groups
zbMATH Keywords: uniformly saturated pro-\(p\) groups
zbMATH Keywords: completed group algebras
zbMATH Keywords: free metabelian pro-\(p\) groups
zbMATH Keywords: free nilpotent groups
zbMATH Keywords: number of generators
zbMATH Keywords: \(p\)-adic analytic groups
author: Vitaliĭ Roman'kov
MaRDI profile type: Publication
cites work: Q3997907
cites work: \(p\)-adic analytic groups
cites work: Automorphism groups of free nilpotent groups
description: scientific article; zbMATH DE number 652183