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: $$a_i^p$$ 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.
 * 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.

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