= Prüfer rank
{c}
{title2=$\operatorname{rk}_{\mathrm P}(G)=\sup_{H\le G,\ H\text{ finitely generated}}d(H)$}
The Prüfer rank bounds the number of <generators of a group> needed for each finitely generated subgroup, without requiring the ambient group to be finitely generated. The additive <group> $\mathbb Q$ has Prüfer rank one: any finitely generated subgroup has a common denominator and is cyclic, but no finite collection generates all rational numbers. Prüfer rank cannot increase under taking subgroups or quotients; for quotients, lift a finite generating set and apply the rank bound to the subgroup generated by its lifts.
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