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Gröbner bases and the computation of group cohomology. (English) Zbl 1050.20036

Lecture Notes in Mathematics 1828. Berlin: Springer (ISBN 3-540-20339-7/pbk). xi, 138 p. (2003).
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.

MSC:

20J06 Cohomology of groups
20-04 Software, source code, etc. for problems pertaining to group theory
20-02 Research exposition (monographs, survey articles) pertaining to group theory
16E05 Syzygies, resolutions, complexes in associative algebras
16Z05 Computational aspects of associative rings (general theory)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation

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