The fundamental theorem of finitely generated abelian groups (or maybe just finite abelian groups) is well-known and can be found in just about any text on the theory of groups or abstract algebra. Any finitely generated abelian group $A$ has a primary decomposition: $$A \simeq \mathbb{Z}^{n}\oplus\bigoplus_{k=1}^{m}\mathbb{Z}_{q_{k}} ,$$ in which the $q_{k}$ are prime powers. It also has an invariant factor decomposition: $$A\simeq \mathbb{Z}^n\oplus\bigoplus_{k=1}^{s}\mathbb{Z}_{f_{k}},$$ in which $f_{1}\mid f_{2}\mid\cdots\mid f_{s}$.
My question is: Who first proved these results?
I looked at several standard texts, and a bunch of online results, but I found only a confused indication of who first established these. Both Robinson and Rotman indicate that the primary decomposition is due to Frobenius-Stickelberger, but might (in part?) date back to Gauss, while some online sources (e.g., MathWorld, usually fairly reliable) call it a Kronecker decomposition, suggesting perhaps a different history. I did not see any indication of who proved the invariant factor decomposition, or which (if not worked out simultaneously) came first.
The book Abelian groups by Fuchs claims that the Fundamental Theorem for Finite Abelain Groups (which Fuchs calls the Basis Theorem) is due to Frobenius and Ludwig Stickelberger (1850-1936, a colleague of Frobenius from Zurich). See also A course in the Theory of Groups by Robinson. However, neither book attributes the more general finitely-generated case to anyone.
The cited paper is the following.
MacTutor claims that this paper gives a proof of the more general structure theorem for finitely generated abelian groups. As I do not speak German, I would struggle to verify this. Therefore, I would tend to believe Fuchs and Robinson, and attribute the MacTutor claim to (a very understandable) human error.