Among finite abelian groups, only the cyclic groups are indecomposable.
As per the title of the question, I am trying to disprove the above statement. On a recent test, I marked this "True" and found that I was incorrect. Still, I am confused and cannot think of a counterexample.
Moreover, take a finite abelian group $G$, then isn't $G$ isomorphic to the direct product of cyclic groups by the Fundamental Theorem for Finite Abelian Groups?
What am I missing here?