On commuting automorphisms of groups
type: Item
instance of: scholarly article
title: On commuting automorphisms of groups
zbMATH Open document ID: 1017.20028
DOI: 10.1007/BF02638378
published in: Archiv der Mathematik
publication date: 2003-03-27T00:00:00Z
review text: Let \(A(G)=\{\alpha\in\Aut(G)\mid x\alpha(x)=\alpha(x)x\) for all \(x\) in \(G\}\) and \(\text{Cent}(G)=\{\alpha\in\Aut(G)\mid\alpha(C(x))=C(x)\) for all \(x\) in \(G\}\). The authors show that \(A(G)\) is not necessarily a subgroup of \(\Aut(G)\) but \(\alpha^2\in\text{Cent}(G)\) for all \(\alpha\in A(G)\) (Lemma 2.4(i)); if \(G\) is Noetherian, then \([G,A(G)]\leq Z_\infty(G)\) (Theorem 1.1); if \(G\) is finite, there is a nilpotent subgroup \(H\) such that the map restricting \(\alpha\in A(G)\) to \(H\) is injective (Theorem 1.2), and \([G^2,A(G)]\leq Z_2(G)\) (Theorem 1.4).
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F45
zbMATH DE Number: 1889271
zbMATH Keywords: commuting automorphisms
zbMATH Keywords: hypercenter
zbMATH Keywords: Engel elements
zbMATH Keywords: nilpotent subgroups
author: Marian Deaconescu
author: Gheorghe Silberberg
author: G. L. Walls
reviewed by: Hermann Heineken
MaRDI profile type: Publication
cites work: Engelsche Elemente Noetherscher Gruppen
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cites work: Q3227605
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full work available at URL: https://doi.org/10.1007/bf02638378
OpenAlex ID: W2001746316
description: scientific article; zbMATH DE number 1889271