On the outer automorphism groups of finitely generated, residually finite groups

Logan, Alan D.

doi:10.1016/j.jalgebra.2014.10.017

Journal of Algebra

2015

DOI: 10.1016/j.jalgebra.2014.10.017

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On the outer automorphism groups of finitely generated, residually finite groups

Alan D. Logan

Abstract: Abstract. Bumagin-Wise posed the question of whether every countable group can be realised as the outer automorphism group of a finitely generated, residually finite group. We give a partial answer to this problem for recursively presentable groups.

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Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

Logan

2019

Advances in Mathematics

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Fix an equilateral triangle group T i = a, b; a i , b i , (ab) i with i ≥ 6 arbitrary. Our main result is: for every presentation P of every countable group Q there exists an HNN-extension T P of T i such that Out(T P ) ∼ = Q. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of "malcharacteristic" subgroups.

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Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

Logan

2019

Advances in Mathematics

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The Bass–Jiang group for automorphism-induced HNN-extensions

Logan

2019

Journal of Algebra

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On the isomorphism problem for split extensions

Snanou¹,

Charkani²

2022

J. Algebra Appl.

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In this paper, we study an important well-known structure operation, namely, the semidirect product (or split extensions) which is a very useful tool to structure certain kinds of groups. More precisely, we study the isomorphism problem for semidirect products and then we determine how isomorphism of semidirect products and conjugacy of the images of the corresponding actions are related. As an application, for two positive integers [Formula: see text] and [Formula: see text], we compute the number of upper isomorphism classes of split extensions of an elementary abelian [Formula: see text]-group of order [Formula: see text] by an elementary abelian [Formula: see text]-group of order [Formula: see text]. Furthermore, we deal with split extensions where the kernel is a non-abelian [Formula: see text]-group of nilpotency class two and the quotient is an elementary abelian [Formula: see text]-group.

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On the outer automorphism groups of finitely generated, residually finite groups

Abstract: Abstract. Bumagin-Wise posed the question of whether every countable group can be realised as the outer automorphism group of a finitely generated, residually finite group. We give a partial answer to this problem for recursively presentable groups.

Cited by 3 publications

References 14 publications

Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

The Bass–Jiang group for automorphism-induced HNN-extensions

On the isomorphism problem for split extensions

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