Free presentation

A free presentation of a group is a quotient $G\cong F/R$ in which $F$ is a free group and $R$ is the normal closure of the chosen relators.

A finite group presentation is balanced when it has the same number of generators and relators.

For a finite group presentation, the relator exponent-sum matrix records in row $i$ the total exponent of each generator in relator $r_i$. It presents the abelianization of the group. For a balanced presentation of a perfect group, it is a unimodular matrix.

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 $F$.

Fox calculus assigns noncommutative derivatives $\partial /\partial x_i$ to words in a free group. The product rule and the identityturn relators into module maps and give the presentation relation sequence.

The Fox derivative of a word in a free group obeys $\partial (uv)/\partial x=\partial u/\partial x+u\partial v/\partial x$ and $\partial x^{-1}/\partial x=-x^{-1}$.