Theorem of structure for abelian groups I was thinking that the structure theorem for abelian groups is not true for the trivial group. Am I missing something or is the theorem just valid for groups of order $&gt;{1}$ ?

Edit: The theorem I'm working with is: Let $G$ be an abelian group of finite order, then there exist integers $d_1,...,d_k$ so that $$G\sim\Bbb{Z}/(d_1)\times...\times\Bbb{Z}/(d_k),$$
and so that $d_i\mid{d_{i+1}}$ for $1\leq{i}\leq{k-1}$. The trivial group clearly doesn't satisfy this.
I'm guessing you're talking about the structure theorem for finitely generated abelian groups, which is a product of groups of some form (depending on the version of the theorem you're talking about.)

The interpretation that works (which is perfectly sound) is that $\{0\}$ is the empty product of abelian groups.
I think you are confusing the trivial group with the empty set...  The latter isn't a group  (no identity )...

The trivial group is cyclic, so satisfies the theorem. ..