Heisenberg group modulo prime

Question

According to Wikipedia, the Heisenberg group modulo $p$, where $p$ is an odd prime, has the presentation $$H(\mathbb{F}_p)=\langle x,y,z\mid x^p=y^p=z^p=1, \ xz=zx, \ yz=zy, \ z=xyx^{-1}y^{-1}\rangle.$$ I could even derive it, but the proof seems to work modulo any integer, not just an odd prime. Why should $p$ be an odd prime? (If it works modulo any integer, it seems a little strange that the Wikipedia article insists on $p$ being an odd prime.)

Answer 1

The Heisenberg group can be defined over any commutative unital ring. It is the group of all uni-upper- triangular matrices over that ring. If the ring is the ring of integers modulo some $n$ you get your group. Traditionally, though, people assume $n=p$ to be prime. Then the group is one of two nonabelian groups of order $p^3$.