Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.246353
Title: A cohomological approach to the classification of $p$-groups
Author: Borge, I. C.
ISNI:       0000 0001 3470 9673
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2001
Availability of Full Text:
Access through EThOS:
Full text unavailable from EThOS. Please try the link below.
Access through Institution:
Abstract:
In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.246353  DOI: Not available
Keywords: Algebraic geometry Mathematics
Share: