Deriving the Discrete Heisenberg Group generators.

Question

How can we derive the generators of the Discrete Heisnberg Group?

Everyone seems to just state this as a given and never actually derive it from scratch.

I'm looking for a (somewhat) elementary derivation

Answer 1

Since the discrete Heisenberg group is defined to be the subgroup of $GL_3(\Bbb{Z})$ consisiting of upper-unitriangular matrices, it is clear that the generators are given by $x,y,z$, where $$ \begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix}=y^bz^cx^a\, $$ see Wikipedia. Here it is enough to consider $x$ and $y$ since $z=[x,y]$ by matrix multiplication.