Computing the Cohomology Ring and Ext-Algebra of Group Algebras

Abstract

This dissertation describes an algorithm and its implementation in the computer algebra system GAP for constructing the cohomology ring and Ext-algebra for certain group algebras kG. We compute in the Morita equivalent basic algebra B of kG and obtain the cohomology ring and Ext-algebra for the group algebra kG up to isomorphism. As this work is from a computational point of view, we consider the cohomology ring and Ext-algebra via projective resolutions.There are two main methods for computing projective resolutions. One method uses linear algebra and the other method uses noncommutative Grobner basis theory. Both methods are implemented in GAP and results in terms of timings are given. To use the noncommutative Grobner basis theory, we have implemented and designed an alternative algorithm to the Buchberger algorithm when given a finite dimensional algebra in terms of a basis consisting of monomials in the generators of the algebra and action of generators on the basis.The group algebras we are mainly concerned with here are for simple groups in characteristic dividing the order of the group. We have computed the Ext-algebra and cohomology ring for a variety of simple groups to a given degree and have thus added many more examples to the few that have thus far been computed.