Absolute central automorphisms that are inner.
type: Item
instance of: scholarly article
title: Absolute central automorphisms that are inner.
zbMATH Open document ID: 1319.20025
author: Mohammad Mehdi Nasrabadi
author: Z. Kaboutari Farimani
published in: Indagationes Mathematicae. New Series
publication date: 2014-11-27T00:00:00Z
review text: For a finite group \(G\), the absolute centre of \(G\) is the subgroup \(L(G)\) of \(G\) left fixed elementwise by all elements of \(\Aut(G)\). By analogy with the group of central automorphisms, the authors consider the group \(\Aut_l(G)\) of those automorphisms of \(G\) which act trivially on \(G/L(G)\). Since clearly \(L(G)\) is contained in the centre \(Z(G)\) of \(G\), every element of \(\Aut_l(G)\) centralizes the group \(\text{Inn}(G)\) of the inner automorphisms of \(G\). The main result of this note is this: if \(G\) is a nonabelian autonilpotent finite \(p\)-group of class 2, then \(\Aut_l(G)=\text{Inn}(G)\) if and only if \(L(G)=Z(G)\) is cyclic.
reviewed by: Marian Deaconescu
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6374452
zbMATH Keywords: absolute central automorphisms
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: inner automorphisms
zbMATH Keywords: absolute centre
MaRDI profile type: Publication
full work available at URL: j.indag.2014.08.003
OpenAlex ID: W2031625524
cites work: The absolute centre of a group
cites work: Endliche Gruppen I
cites work: On the minimal number of generators of finite non-abelian p-groups having an abelian automorphism group
cites work: Q4317890
DOI: 10.1016/J.INDAG.2014.08.003
description: scientific article; zbMATH DE number 6374452