The conjugacy problem and groups of automorphisms

共轭问题和自同构群

• 批准号: 535115960
• 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: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/535115960?language=en
• 关键词: conjugacy; problem; groups; automorphisms

The conjugacy problem is one of the fundamental problems in group theory. It was stated in 1911 by Dehn as one of three major problems along with the word problem and the isomorphism problem. In recent years it became very popular due to the AAG cryptosystem that requires groups with hard conjugacy problem. The investigation of certain types of groups and their conjugacy problem has therefore become an important task in group theory. In this project the conjugacy problem is investigated in three types of groups: subgroups of integral matrix groups, automorphism groups of torsion free finitely generated nilpotent groups and automorphism groups of finitely generated free groups. The conjugacy problem in the full general linear group over the integers GL(n,Z) was solved by Grunewald (1980) and practical methods for this problem were developed by Eick, Hofmann and O'Brien (2019) and by Bley, Hofmann and Johnston (2022). The conjugacy problem for subgroups of GL(n,Z) is open and it is our plan to consider this for certain cases. Torsion free finitely generated nilpotent groups have a well developed algorithmic theory and there are effective methods available to solve their conjugacy problem. However, their automorphism groups are a different case and their conjugacy problem is wide open. We will consider different approaches towards solving this conjugacy problem. Finitely generated free groups have been investigated for a long time and their conjugacy problem can be solved readily. Again, this is very different for their automorphism groups. Bogopolski (1889) has described a solution for automorphism groups of free groups on two generators. It is our aim to consider this problem further.

共轭问题是群论中的基本问题之一。1911年，Dehn将其与词问题和同构问题并列为三大问题之一。近年来，由于AAG密码体制要求群具有难共轭问题，它变得非常流行。因此，研究某些类型的群及其共轭问题已成为群论中的一个重要任务。本文研究了三类群的共轭问题：整矩阵群的子群、无挠有限生成幂零群的自同构群和有限生成自由群的自同构群。Grunewald(1980)解决了整数GL(n，Z)上全一般线性群的共轭问题，Eick，Hofmann和O‘Brien(2019)以及Bley，Hofmann和Johnston(2022)发展了解决这一问题的实用方法。GL(n，Z)的子群的共轭问题是公开的，我们计划在某些情况下考虑这一问题。无挠有限生成的幂零群有很好的算法理论，并且有有效的方法来解决它们的共轭问题。然而，它们的自同构群是另一种情况，它们的共轭问题是完全公开的。我们将考虑不同的方法来解决这个共轭问题。有限生成自由群的研究由来已久，它们的共轭问题可以很容易地解决。同样，这对于他们的自同构群来说是非常不同的。Bogopolski(1889)描述了两个生成元上自由群的自同构群的一个解决方案。我们的目标是进一步考虑这个问题。