Every group is an outer automorphism group of a finitely generated group.
type: Item
instance of: scholarly article
title: Every group is an outer automorphism group of a finitely generated group.
zbMATH Open document ID: 1082.20021
author: Inna Bumagin
author: Daniel T. Wise
published in: Journal of Pure and Applied Algebra
publication date: 2005-06-23T00:00:00Z
review text: Due to the restriction to finitely generated groups, the title of the paper is slightly misleading: To be correct, it should begin with `Every countable group \dots'. Without the restriction to finitely generated groups it can be shown that any group is the outer automorphism group of some group. In fact the underlying groups can come from particular classes of groups as shown in recent papers and discussed in the introduction of this paper. An interesting application of small cancellation theory provides for every countable group \(Q\) the existence of a special finitely generated group \(N\) with \(\text{Out}(N)\cong Q\). The authors use a variant of \textit{E. Rips}' construction [Bull. Lond. Math. Soc. 14, 45-47 (1982; Zbl 0481.20020)].
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20F06
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 2182121
zbMATH Keywords: outer automorphism groups
zbMATH Keywords: small cancellation groups
zbMATH Keywords: finitely generated groups
zbMATH Keywords: finitely presented groups
zbMATH Keywords: residually finite groups
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.12.033
OpenAlex ID: W1992897714
cites work: Automorphism Groups of Residually Finite Groups
cites work: ALL GROUPS ARE OUTER AUTOMORPHISM GROUPS OF SIMPLE GROUPS
cites work: On locally finite \(p\)-groups and a problem of Philip Hall's
cites work: Outer automorphism groups of metabelian groups
cites work: On Dehn's algorithms for the conjugacy and word problems, with applications
cites work: A class of groups all of whose elements have trivial centralizers
cites work: Q5552195
cites work: Isometry transformations of hyperbolic 3-manifolds
cites work: On Powers of Elements in S-Groups
cites work: Q4145882
cites work: Any group is represented by an outerautomorphism group
cites work: Subgroups of small Cancellation Groups
cites work: Topology of finite graphs
cites work: Q4041788
cites work: Residual Finiteness of Quasi-Positive One-Relator Groups
cites work: A RESIDUALLY FINITE VERSION OF RIPS'S CONSTRUCTION
DOI: 10.1016/J.JPAA.2004.12.033
description: scientific article; zbMATH DE number 2182121