Gröbner bases and the computation of group cohomology This monograph develops the Gröbner basis methods needed to perform efficient state of the art calculations in the cohomology of finite groups. Results obtained include the first counterexample to the conjecture that the ideal of essential classes squares to zero. The context is J. F. Carlsons minimal resolutions approach to cohomology computations. TOC:Introduction. Part I Constructing minimal resolutions. 1. Bases for finite dimensional algebras and modules. 2. The Buchberger Algorithm for modules. 3. Constructing minimal resolutions. Part II Cohomology ring structure. 4. Gröbner bases for graded commutative algebras. 5. The visible ring structure. 6. The completeness of the presentation. Part III Experimental results. 7. Experimental results. A. Sample cohomology calculations. Epilogue. References. Index