Indecomposable finite abelian groups
= Indecomposable finite abelian groups
A finite abelian group is indecomposable exactly when it is cyclic of prime-power order. The structure theorem proves necessity. Conversely, every nonzero subgroup of $C_{p^n}$ contains its unique subgroup of order $p$, so two nonzero subgroups cannot be complementary direct summands.