TITLE: Who first proved the fundamental theorem of finitely generated (or finite) abelian groups?
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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.

G. Frobenius, L. Stickelberger, Uber Grubben von vertauschbaren Elementen, J. reine u. angew. Math., 86 (1878), 217-262.

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.