Homological Methods in Algebra

In this thesis, we apply homological methods to the study of groups in two ways: firstly, we generalise the results of [12] to a more general class of categories than posets, including finite groups which satisfy a particular
cohomological condition. We then show that the only finite group satisfying this condition is the trivial group, but our results still hold in more generality than the originals, and we suggest a path to further generalisation.
Secondly, we study the representation theory of certain groups by passing their actions on certain simplicial complexes to actions on the homologies of those complexes.