I'm a little confused about the definition of the rank of a finitely-generated abelian group $G$, which I've read is the size of a largest linearly independent set in $G$. I'm looking at a proof that proves that a certain abelian group has rank $n$ by exhibiting a linearly independent set of $n$ elements and showing that it's maximal. But how is this enough? Just because that particular independent set can't be extended doesn't on its own stop there from being some totally different linearly independent set with more elements.
I'm led to infer that all maximal independent sets in a finitely-generated abelian group have the same number of elements, but this isn't at all obvious to me. Why is it true? (I'd like to be able to see this without using tensors---I'm aware the arguments are more succinct from that perspective but I haven't learned enough about tensors yet.)
Edit: Just realized I accidentally assumed the groups were free - I meant to assume just that they’re finitely generated.