Gröbner bases and the computation of group cohomology.
The computation of the cohomology ring of finite groups is in general a difficult task; two major obstacles are found right away: how to find appropriate resolutions and how to determine whether we have found all the existing relations. In this book, the author explains how to overcome the above hurdles and the implementation of computer programs that perform the necessary computations to obtain complete descriptions of the cohomology ring of finite \(p\)-groups. Here are the steps carried out by the author: first he describes a method to construct a minimal resolution for the \(kG\)-module \(k\) (\(k\) is a field of characteristic \(p\)), a major ingredient in this step is to use Gröbner bases to determine a minimal set of generators for the kernel of a \(kG\)-module map. The author describes both the theoretical foundations of the algorithms and the implementations of them in the computer. The second step is to determine the product in the cohomology and a full set of relations, the key ingredient in this step is the implementation of a criterion found by J. Carlson, again the author describes in great detail both the foundation of the algorithm and the implementation in the computer. The author closes with several examples of cohomology rings providing, among other things, a presentation, restriction to subgroups and essential classes. Finally, the author refers to his www-page for more details, programs and full packages used in this book.
scientific article; zbMATH DE number 2012371