On automorphisms of \(A\)-groups.
type: Item
instance of: scholarly article
title: On automorphisms of \(A\)-groups.
zbMATH Open document ID: 1178.20020
author: Martin R. Pettet
published in: Archiv der Mathematik
publication date: 2009-01-30T00:00:00Z
review text: A group \(G\) is said to be an \(A\)-group if \(xx^\alpha=x^\alpha x\) for all \(a\in G\) and \(\alpha\in\Aut(G)\). Let \(A_C(G)\) denote the subgroup of \(\Aut(G)\) consisting of all automorphisms that leave invariant the centralizer of each element of \(G\). By \(\Aut_c(G)\) we denote the group of all central automorphisms of \(G\). The quotient \(\Aut(G)/\Aut_c(G)\) has exponent at most \(2\). The main result of this interesting paper is the following Theorem 1.1. Let \(r\) be a positive integer and \(p>2\) a prime. Then there exist infinitely many finite \(p\)-groups \(G\) such that \(G\) is an \(A\)-group, \(A_C(G)=\Aut_c(G)\) and the elementary Abelian \(2\)-group \(\Aut(G)/\Aut_c(G)\) has order \(2^r\).
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5500767
zbMATH Keywords: \(A\)-groups
zbMATH Keywords: graphs
zbMATH Keywords: automorphisms
zbMATH Keywords: automorphism groups
zbMATH Keywords: centralizers
zbMATH Keywords: central automorphisms
zbMATH Keywords: finite \(p\)-groups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s00013-008-2807-5
OpenAlex ID: W1998113470
DOI: 10.1007/S00013-008-2807-5
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description: scientific article; zbMATH DE number 5500767