= Solution
By the finite case of the structure theorem, a finite abelian <group> is a direct sum of cyclic <groups> of prime-power order. If it is indecomposable, this decomposition can have only one nonzero summand, so the <group> is cyclic of order $p^n$ for some prime $p$.
Conversely, every nontrivial <subgroup> of the cyclic <group> $C_{p^n}$ contains its unique <subgroup> of order $p$. Hence two nontrivial <subgroups> cannot have trivial intersection. They therefore cannot be the two summands of an internal direct sum. Thus the <indecomposable finite abelian groups> are precisely
$$
\boxed{C_{p^n}\quad(n\geq1)}.
$$
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