The conjugacy problem and groups of automorphisms

共轭问题是群论中的基本问题之一。这是在1911年由德恩作为一个三大问题之沿着随着字的问题和同构问题。近年来，由于AAG密码系统要求群具有硬共轭问题，它变得非常流行。因此，研究某些类型的群及其共轭性问题已成为群论中的一个重要课题。本文研究了三类群的共轭性问题：整矩阵群的子群，挠自由群的自同构群，挠自由群的自同构群。整数GL（n，Z）上的全一般线性群中的共轭问题由格鲁内瓦尔德（1980）解决，Eick，Hofmann和奥布莱恩（2019）以及Bley，Hofmann和约翰斯顿（2022）开发了这个问题的实用方法。GL（n，Z）的子群的共轭问题是开放的，我们计划在某些情况下考虑这一点。挠自由群生成的幂零群的算法理论已经很成熟，并且有有效的方法来解决其共轭性问题。然而，他们的自同构群是一个不同的情况下，他们的共轭问题是开放的。我们将考虑解决这个共轭问题的不同方法。生成自由群的共轭性问题是一个很容易解决的问题。同样，对于它们的自同构群，这是非常不同的。Bogopolski（1889）描述了两个生成元上的自由群的自同构群的解。我们的目标是进一步考虑这个问题。

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.