Topics in the theory of soluble groups of finite rank

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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ORCID for Peter Kropholler: orcid.org/0000-0001-5460-1512

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