I've been having trouble finding resources because searching up "rank" for abelian groups always brings up the free abelian rank, which is not what I'm looking for.
However I'm interested in the more general concept of group rank. Define $${\rm rank}(G)=\min\{ \#S | S \text{ generates }G\}$$
I'd like to know if there's a reference/theorem regarding the rank (in the sense above) of finitely generated abelian groups with torsion elements.
- Is it true that the rank of such a group is the sum of the free rank and the rank of the torsion group?
- How subgroups of finitely generated abelian groups are characterized?
- In particular, what is the rank of $\mathbb{Z}^n \oplus \mathbb{Z}/m\mathbb{Z}$ and what are its subgroups?
For torsion free and finitely generated abelian groups, say, $A=\mathbb{Z}^n$, there are many references that show that its subgroups are isomorphic to $\mathbb{Z}^r$ for $r\leq n$. How much can be generalized in those claims to case with torsion?
I'm sure this is standard material, and my only issue is not finding when looking this up, as other concepts appear instead. Any references appreciated.