Finitely presented group

Remarks

The pair $(S, R)$ is a finite presentation of $G$. Example: $\Z^2 \cong \langle a, b \mid a b a^{-1} b^{-1} \rangle$; the map $a \mapsto (1, 0)$, $b \mapsto (0, 1)$ induces the isomorphism, since the quotient is abelian and generated by two elements, and $\Z^2$ is the free abelian group on them. Every finite group $G$ is finitely presented, with $S = G$ and $R = \set{ g h k^{-1} : g, h, k \in G, \ gh = k \text{ in } G}$. Non-example: the additive group $\Q$ is not finitely presented; it fails already because it is not finitely generated, since a subgroup generated by finitely many fractions has bounded denominators. A finitely generated group need not be finitely presented; the lamplighter group $\Z/2 \wr \Z$ is such an example. If $G$ is finitely presented, then for every finite generating set $S$ of $G$ there is a finite set of relations $R$ with $G \cong \langle S \mid R \rangle$.