There are places (including on MSE) online where one may easily find modern proofs of the fundamental theorem of finitely generated Abelian groups. Wikipedia, for instance, mentions a seemingly interesting proof by Poincare
The fundamental theorem for finitely generated abelian groups was proven by Henri Poincaré in (Poincaré 1900), using a matrix proof (which generalizes to principal ideal domains). This was done in the context of computing the homology of a complex, specifically the Betti number and torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part.
Now, I'm not familiar with a ton of the vast study of topological groups, but know a reasonable amount of some basic ideas in groups, representations, topology, and some homology, so I think this proof would be accessible to me, but I can't seem to find it spelled out in any detail online. Stillwell
briefly outlines the proof, but I'd really like to see more explicitly the computation of Betty numbers and Torsion coefficients performed by Poincare, so I was hoping to get a reference for this proof somewhere.
Best Answer
Munkres "Elements of Algebraic Topology" does the proof in detail in the way that you're asking.
The Fundamental Theorem of Finitely Generated Abelian Groups is Theorem 4.2, Section 4, Chapter 1. See discussion thereafter and then the proof of the theorem is given in Section 11, Chapter 1 (after introducing the machinery listed in your quote).
I'd imagine that other books on Algebraic Topology might also cover this way of proving the fundamental theorem, if you end up not liking the exact presentation you find in Munkres.