Wharton, Elizabeth (2006) *The model theory of certain infinite soluble groups.* PhD thesis, University of Oxford.

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## Abstract

This thesis is concerned with aspects of the model theory of infinite soluble groups. The results proved lie on the border between group theory and model theory: the questions asked are of a model-theoretic nature but the techniques used are mainly group-theoretic in character.

We present a characterization of those groups contained in the universal closure of a restricted wreath product U wr G, where U is an abelian group of zero or finite square-free exponent and G is a torsion-free soluble group with a bound on the class of its nilpotent subgroups. For certain choices of G we are able to use this characterization to prove further results about these groups; in particular, results related to the decidability of their universal theories.

The latter part of this work consists of a number of independent but related topics.

We show that if G is a finitely generated abelian by-metanilpotent group and H is elementarily equivalent to G then the subgroups and are elementarily equivalent, as are the quotient groups and .

We go on to consider those groups universally equivalent to , where the free groups of the variety are residually finite p-groups for infinitely many primes p, distinguishing between the cases when c = 1 and when c > 2.

Finally, we address some important questions concerning the theories of free groups in product varieties , where is a nilpotent variety whose free groups are torsion-free; in particular we address questions about the decidability of the elementary and universal theories of such groups. Results mentioned in both of the previous two

paragraphs have applications here.

Item Type: | Thesis (PhD) |
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Subjects: | H - N > Mathematical logic and foundations D - G > Group theory and generalizations |

Research Groups: | Algebra Research Group Mathematical Logic Group |

ID Code: | 1461 |

Deposited By: | Eprints Administrator |

Deposited On: | 02 Feb 2012 16:19 |

Last Modified: | 02 Feb 2012 16:19 |

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