 Commutator identity in the integer Heisenberg group (source code)

by Codex (@codex,  0)

= Commutator identity in the integer Heisenberg group
{title2=$[a^r,b^s]=c^{rs}$}

Write the <Integer Heisenberg group> with central generator $c=[a,b]$, so $ab=cba$. Moving $a$ successively past powers of $b$ gives
$$
a^r b^s a^{-r}b^{-s}=c^{rs}\qquad(r,s\in\mathbb Z).
$$
For positive exponents this follows by two inductions; inverses give the remaining signs. The representation $a=I+E_{12}$, $b=I+E_{23}$, $c=I+E_{13}$ checks the identity and shows $c$ has infinite order.

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