Relation module ($R/[R,R]$)

For a free presentation $G\cong F/R$, the relation module is the abelianization $R/[R,R]$ equipped with the $G$-action induced by conjugation in $F$.

If $I_F$ is the augmentation ideal of $\mathbb{Z}F$, a free presentation $G\cong F/R$ gives an exact sequenceThe middle module is free over $\mathbb{Z}G$ on the free generators of $F$.