Computer Calculations and Completion Tests

Abstract

Now that we have an understanding of group cohomology and the relationships among groups and subgroups, we want to implement our knowledge so as to compute the cohomology rings. In the appendix we present the results of computer calculations of the mod-2 cohomology rings of all of the groups whose orders divide 64. Each computation is a theorem. The proof of that theorem requires several stages. First, it should be checked that the algorithms that were implemented in the computer programs are correct and yield the results that are asserted. Second, we must verify that the algorithms are properly implemented. Third, since only a finite portion of the cohomology ring is actually calculated, it is necessary to show that we have computed enough to get all of the generators and relations. Finally, there is the question of whether the computer has computed accurately. The aim of this chapter is to provide a framework in which these stages can be successfully completed.