# Power-commutator presentation

A power-commutator presentation of a group $G$ for a prime number $p$ 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: $a_i^p$ is written as a product of powers of $a_k, k > i$, with the $k$s in increasing order as we go from left to right in the product. The exponent on $a_k$ for $k > i$ is denoted $\beta(i,k)$.
• 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. The exponent of $a_k$ for $k > \max \{ i , j \}$ is denoted $\beta(i,j,k)$.
For a group of prime power order $p^n$, a power-commutator presentation is termed consistent if it uses exactly $n$ generators.