The conjugacy problem and groups of automorphisms

点击翻译按钮获取中文摘要

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.