The conjugacy problem and canonical representatives in finitely generated nilpotent groups

Abstract

We introduce a variation on the conjugacy problem for elements and subgroups in a finitely generated nilpotent group G given by a nilpotent presentation and we describe effective algorithms for its solution. While the classical conjugacy problem takes elements or subgroups a and b of G and asks to construct $g\in G$ with $a^g=b$, our variation defines and determines a canonical representative $Can{o}_G\left(a\right)$ in $a^G$. This allows to solve the conjugacy problem via an equality test $Can{o}_G\left(a\right)=Can{o}_G\left(b\right)$. Additionally, our algorithms compute the associated centralizers or normalizers, respectively. We exhibit a variety of examples to demonstrate that our new methods are highly effective and often outperform the existing methods to solve the conjugacy problems for elements and subgroups in finitely generated nilpotent groups.