Relators and relations; finitely generated, finitely related, and finite presentations

In a presentation $\langle X\mid R\rangle$ as in Group presentation by generators and relations, an element $r\in R\subseteq F(X)$ is called a defining relator. The equation $r=1$ that it imposes in the quotient is a defining relation. More generally, an equation $u=v$ may be recorded by the relator $u^{-1}v$. The published definition uses the common looser convention of calling the members of $R$ relations; both conventions define the same quotient group.

A presentation is finitely generated when $X$ is finite, finitely related when $R$ is finite, and finite when both $X$ and $R$ are finite. A group is called finitely generated, finitely related, or finitely presented when it admits a presentation with the corresponding property. For finitely generated groups this agrees with generation by a finite subset in the sense of The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups.