Characterizing inner automorphisms of groups
type: Item
instance of: scholarly article
title: Characterizing inner automorphisms of groups
zbMATH Open document ID: 0683.20025
DOI: 10.1007/BF01190261
author: Martin R. Pettet
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: P. E. Schupp has shown, using small cancellation theory, that the inner automorphisms of a group G are characterized by the fact that they extend to any group which contains G as a subgroup. Here we record an elementary proof of Schupp's result based on a graph construction and an associated group presentation. An advantage of this approach is that it yields also the dual result that the inner automorphisms of G are precisely those automorphisms which can be lifted to any group of which G is a homomorphic image. In addition, both characterizations are seen to apply within certain restricted classes of groups (such as the class of p- groups of a given infinite order). Much of the argument is devoted to establishing a general version of a well-known theorem of H. Heineken and H. Liebeck.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 4118590
zbMATH Keywords: inner automorphisms
zbMATH Keywords: presentation
zbMATH Keywords: homomorphic image
MaRDI profile type: Publication
cites work: Q5771659
cites work: The occurrence of finite groups in the automorphism group of nilpotent groups of class 2
cites work: Q5795193
cites work: On Inner Automorphisms of Finite Groups
cites work: Graphs with given infinite group
cites work: A Characterization of Inner Automorphisms
cites work: The occurrence of groups as automorphisms of nilpotent p-groups
full work available at URL: https://doi.org/10.1007/bf01190261
OpenAlex ID: W2038665299
description: scientific article; zbMATH DE number 4118590