# nilpotent subgroups of automorphism group

Where can I read about nilpotent subgroups of $Aut(Z^{n})$? ($Z$ is an infinite cyclic group). I will be thankful for any referalls.

• I think this group is better known as $\mathrm{GL}(n,\mathbb Z)$. – Matt Samuel Mar 25 '17 at 18:45

Nilpotent subgroups of $Aut(\mathbb{Z}^n)=GL_n(\mathbb{Z})$ are given, for example, by the integer Heisenberg groups $H_{2m+1}$. By a Theorem of Swan, every polycyclic group can be embedded into some $GL_n(\mathbb{Z})$, so every nilpotent polycyclic group is an example. For a crystallographic group $\Gamma$ and its point group $G$ we have a short exact sequence $$1\rightarrow \mathbb{Z}^n\rightarrow \Gamma\rightarrow G\rightarrow 1,$$ which defines the holonomy representation $$h_{\Gamma}:G\rightarrow Aut(\mathbb{Z}^n)=GL_n(\mathbb{Z}).$$