Power-commutator presentation

Definition

A power-commutator presentation of a group $G$ is a presentation with generating set $a_i,i\in I$ for a totally ordered indexing set $I$ and relations of the form:

• power relations: $a_i^{r_i}$ where $r_i$ is a nonnegative integer depending on $i$.
• commutator relations: The commutator $[a_i,a_j]$ is written as a product of powers of $a_k,k>max\{i,j\}$, with the $k$s in increasing order as we go from left to right in the product.

Facts

• Every group of prime power order has a power-commutator presentation