Applications of cohomology in group theory and number theory

群扩展的构造和分类在群论的许多应用中起着核心作用。计算群理论为构造初等阿贝尔模的强同构扩展提供了有效的方法。这种对模块的限制极大地限制了可能的应用程序。我们的目的是利用上同调理论发展一种新的有效方法来构造以任意有限群为模的强同构扩展。然后我们将研究这种方法的变体。例如，我们想开发一种有效的方法来构造同构的所有扩展，其中模块作为拟合子群或作为派生或下中心级数的项嵌入。此外，我们计划将我们的新算法应用于群论和数论。首先，我们希望将Small Groups库扩展到最多10,000个订单的所有组，订单上很少有例外。然后，我们研究了具有给定派生子群和派生商的有限亚丫群的构造到同构。这些群在数论中扮演着重要的角色，因为它们与数域的非分支扩展的伽罗瓦群有关。

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.