Prüfer rank (${\mathrm{rk}}_{\mathrm{P}}(G)={\sup }_{H\le G,H\text{ finitely generated}}d(H)$)

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.

For a normal subgroup $N$ with Prüfer rank $r$ and quotient of rank $s$, let $H=⟨h_1,\dots ,h_k⟩\le G$. Choose at most $s$ elements $x_j\in H$ generating the image of $H$. Write $h_i=w_i(x_1,\dots ,x_s)n_i$ with $n_i\in N\cap H$. The finitely generated subgroup $⟨n_1,\dots ,n_k⟩$ needs at most $r$ generators, so $H$ needs at most $r+s$. The full intersection $H\cap N$ need not be finitely generated.