Generators and relations:

Denoting by $B$ the set of generators $a_1, a_2,\dots, a_k$ of $F_k$, any collection $S$ of words

$$a_{i_1}^{n_1}a_{i_2}^{n_2}\dots, a_{i_p}^{n_p},\quad a_{i_j}\in B,\;n_j \in \mathbb{Z}, \; 1\leq j \leq p. \eqno(23.2)$$

gives rise to a group $F_k/\langle S\rangle$ where $\langle S\rangle$ denotes the normal subgroup generated by $S$. Conversely, let $H$ be a finitely generated group and $\phi$ be as in the theorem. We take a set $R$ of words (23.2) generating the kernel of $\phi$ and write

$$H = F_k/\langle R\rangle. \eqno(23.3)$$

The elements of $R$ are called relators and the set of equations

$$a_{i_1}^{n_1}a_{i_2}^{n_2}\dots, a_{i_p}^{n_p} = 1 \eqno(21.4)$$

obtained by setting each relator to $1$ are called the relations for the group with respect to $\phi$. The list of generators $\{a_1, a_2,\dots, a_k\}$ and relations among them uniquely specifies $H$ through equation (23.3). If a relation in the list (23.4) is a consequence of others, for example if one of them is the product of two others, we may clearly drop it from the list thereby shortening the list. In practice one would try to keep the list of relations down to a minimum. Such a description of $H$ is called a presentation of the group $H$ through generators and relations. A group in general has many presentations and it is usually very difficult to decide whether or not two presentations represent the same group.