Applications of cohomology in group theory and number theory

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Constructions and classifications of group extensions play a central role in many applications of group theory. Computational group theory provides effective methods to construct up to strong isomorphism extensions with an elementary abelian module. This restriction on the module limits the possible applications significantly. Our aim is to use cohomology theory in the development of a new effective method to construct up to strong isomorphism extensions with an arbitrary finite group as module. Then we will investigate variations of this method. For example, we want to develop an effective method to construct up to isomorphism all extensions in which the module embeds as the Fitting subgroup or as a term of the derived or lower central series. Further, we plan to apply our new algorithms in group and number theory. First, we want to extend the Small Groups Library to all groups of the orders at most 10.000 with few exceptions on the orders. Then, we want to investigate the construction up to isomorphism of the finite metabelian groups with a given derived subgroup and derived quotient. These groups play a role in number theory, as they are related to the Galois groups of unramified extensions of number fields.