Vittis, Jason Marcus;
(2019)
The D(2)-Problem for some metacyclic groups.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
We study problems relating to the D(2)-problem for metacyclic groups of type G(p,p-1) for p an odd prime. Specifically we build on J Nadim’s work, which showed that over the integral group ring of G(5,4), the module of integers (upon which the group acts trivially) admits a diagonal resolution and a minimal representative for the third syzygy is R(2)+[y-1). Motivated by this result, we show that R(2)+[y-1) is both full and straight over the integral group ring of G(p,p-1) where p is any odd prime. Given FEA Johnson’s work on the D(2)-problem, this immediately leads to the conclusion that G(5,4) satisfies the D(2)-property, as well as providing a sufficient condition for the D(2)-property to hold for G(p,p-1), namely the condition that R(2)+[y-1) is a minimal representative for the third syzygy. Following this result, we prove a theorem, which in tandem with work from FEA Johnson relating to the module R(2) significantly simplifies the calculations required to show that R(2)+[y-1) is a minimal representative for the third syzygy. Finally, we carry out these calculations for the group G(7,6) and prove that G(7,6) satisfies the D(2)-property.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | The D(2)-Problem for some metacyclic groups |
| Event: | UCL |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2019. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10088009 |
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