The cohomology of a finite matrix quotient group

Abstract

In this work, we find the module structure of the cohomology of the group of four by four upper triangular matrices (with ones on the diagonal) with entries from the field on three elements modulo its center. Some of the relations amongst the generators for the cohomology ring are also given. This cohomology is found by considering a certain split extension. We show that the associated Lyndon-Hochschild-Serre spectral sequence collapses at the second page by illustrating a set of generators for the cohomology ring from generating elements of the second page. We also consider two other extensions using more traditional techniques. In the first we introduce some new results giving degree four and five differentials in spectral sequences associated to extensions of a general class of groups and apply these to both the extensions.