Applications of cohomology in group theory and number theory

上同调在群论和数论中的应用

• 批准号: 171126687
• 负责人: Professorin Dr. Bettina Eick
• 金额: --
• 依托单位: Institut für Analysis und Algebra
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171126687?language=en
• 关键词: Applications; cohomology; group; theory; number

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.

群扩张的构造和分类在群论的许多应用中起着核心作用。计算群论提供了有效的方法来构造与初等交换模的强同构扩张。对模块的这种限制极大地限制了可能的应用。我们的目标是利用上同调理论，发展一种新的有效的方法来构造强同构扩展与任意有限群作为模块。然后我们将研究这种方法的变化。例如，我们想开发一种有效的方法来构造同构的所有扩展，其中模块嵌入为Fitting子群或作为导出或下中心级数的一项。此外，我们计划将我们的新算法应用于群论和数论。首先，我们希望将小型组库扩展到订单的所有组，最多10.000个订单，只有少数订单例外。然后，我们研究了具有给定导子群和导商的有限亚阿贝尔群的同构构造。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歧扩张的伽罗瓦群有关。