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Topics in the theory of soluble groups of finite rank

Topics in the theory of soluble groups of finite rank
Topics in the theory of soluble groups of finite rank
This thesis contains a spectrum of different results all of which, broadly speaking, are motivated by the structure of soluble groups obeying various finiteness conditions. Chapter 1 contains introductory material required throughout the thesis. In chapters 2 and 3, we study endomorphisms of nilpotent groups of finite rank and find criteria to guarantee that they are automorphisms, generalising (independent) work of Farkas and Wehrfritz. Chapter 4 exploits the Mal’cev correspondence for divisible nilpotent groups to characterise so-called powered nilpotent groups, and also contains refinements of results due to Segal. Chapter 5 contains an explicit construction of the free Lie algebra on a module, along with an exposition of the theory of algebraic theories and functors. Finally, in chapter 6 we give an explicit characterisation of the socle series of certain modules over the class of commutative Von Neumann regular rings, confirming conjectures of Usher.
University of Southampton
Durham, Hector
306f01c2-48b4-4476-a3d5-3ebb968ec986
Durham, Hector
306f01c2-48b4-4476-a3d5-3ebb968ec986
Kropholler, Peter
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4

Durham, Hector (2018) Topics in the theory of soluble groups of finite rank. University of Southampton, Doctoral Thesis, 85pp.

Record type: Thesis (Doctoral)

Abstract

This thesis contains a spectrum of different results all of which, broadly speaking, are motivated by the structure of soluble groups obeying various finiteness conditions. Chapter 1 contains introductory material required throughout the thesis. In chapters 2 and 3, we study endomorphisms of nilpotent groups of finite rank and find criteria to guarantee that they are automorphisms, generalising (independent) work of Farkas and Wehrfritz. Chapter 4 exploits the Mal’cev correspondence for divisible nilpotent groups to characterise so-called powered nilpotent groups, and also contains refinements of results due to Segal. Chapter 5 contains an explicit construction of the free Lie algebra on a module, along with an exposition of the theory of algebraic theories and functors. Finally, in chapter 6 we give an explicit characterisation of the socle series of certain modules over the class of commutative Von Neumann regular rings, confirming conjectures of Usher.

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Submitted date: June 2018

Identifiers

Local EPrints ID: 457803
URI: http://eprints.soton.ac.uk/id/eprint/457803
PURE UUID: d5ef4b6e-ae1e-4ee0-8798-a16adeea52a8
ORCID for Peter Kropholler: ORCID iD orcid.org/0000-0001-5460-1512

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Date deposited: 16 Jun 2022 17:05
Last modified: 17 Mar 2024 03:31

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Contributors

Author: Hector Durham
Thesis advisor: Peter Kropholler ORCID iD

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