The goal of this paper is to explore the first cohomology group of groups G that are not necessarily finitely generated. Our focus is on l^p-cohomology, 1 leq p leq infty, and what results regarding finitely generated groups change when G is infinitely generated. In particular, for abelian groups and locally finite groups, the l^p-cohomology is non-zero when G is countable, but vanishes when G has sufficient cardinality. We then show that the l^infty-cohomology remains unchanged for many classes of groups, before looking at several results regarding the injectivity of induced maps from embeddings of G-modules. We present several new results for countable groups, and discuss which results fail to hold in the general uncountable case. Lastly, we present results regarding reduced cohomology, including a useful lemma extending vanishing results for finitely generated groups to the infinitely generated case. / Ph. D. / The goal of this paper is to use a technique that originated in algebraic topology to study the properties of a structure called a group. Groups are collections of objects that interact with each other through an operation that obeys certain properties. Groups arise when considering many different mathematical questions, and they were first studied when looking at the different symmetries an object can have. Classifying the different properties of a group is an active area of mathematical research. We seek to do this by looking at collections of maps from a particular group to the real or complex numbers, then studying how the group shifts these functions.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/77517 |
| Date | 25 April 2017 |
| Creators | Eastridge, Samuel Vance |
| Contributors | Mathematics, Linnell, Peter A., Mihalcea, Constantin Leonardo, Ball, Joseph A., Rossi, John F. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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