Power-commutator presentation

Definition

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.

Facts

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