# Deriving the Discrete Heisenberg Group generators.

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

• The Wikipedia page gives an explicit formula for an arbitrary element in terms of the generators x and y (z can be written in terms of x and y also, Wikipedia gives the computation for that). This is under the section "discrete Heisenberg group". You can try to verify these formulas by computation. – Lorenzo Dec 18 '18 at 15:52
• I have no idea how i missed that, but still a derivation from scratch would be nice – user371732 Dec 18 '18 at 16:13

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.