Prüfer rank

《Lecture Notes on Bayesian Estimation and Classification》 · 第 99 页 · Section 4.2「Direct Sums of Cyclic and Quasicyclic Groups」

If $p$ is a prime and $G$ an abelian group, the $p$-rank of $G$, $r_p(G)$, is defined as the cardinality of a maximal independent subset of elements of $p$-power order. Similarly the $0$-rank or torsion-free rank $r_0(G)$ is the cardinality of a maximal independent subset of elements of infinite order. Also important is the Prüfer rank, often just called the rank of $G$, $r(G)=r_0(G)+\max_p r_p(G)$.

《Lecture Notes on Bayesian Estimation and Classification》 · 第 422 页 · Section 14.1「Finitely Generated Groups and Finitely Presented Groups」，Exercise 14.1.3 定义段

A group $G$ has finite Prüfer rank $r$ if every finitely generated subgroup can be generated by $r$ elements and $r$ is the least such integer.