On the Dimension of Matrix Representations of Finitely Generated Torsion Free Nilpotent Groups

It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into GLn(Z) for an appropriate n ∈ N; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in [6]. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented in [6]. 1. Background Information In this section we will review basic facts about polycyclic and nilpotent groups. We refer the reader to [3] and [2] for more information on these groups. 1.1. Polycyclic Groups. Definition 1. [3] A group is called polycyclic if it admits a finite subnormal series G = G1 ⊲ G2 ⊲ G3 ⊲ · · · ⊲ Gn+1 = 1 where each Gi/Gi+1 is cyclic. The number of infinite factors in the polycyclic series is called the Hirsch length, and is independent of the polycyclic series chosen. Since each factor is cyclic there exists an xi ∈ G such that 〈xiGi+1〉 = Gi/Gi+1. We call the sequence X = [x1, x2, · · · , xn] a polycyclic sequence for G. The sequence of relative orders of X is the sequence R(X) = (r1, · · · , rn) where ri = [Gi : Gi+1]. We denote the set of indices in which ri is finite by I(X). Polycyclic groups have finite presentation