Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 1E / iii / 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

Solved by gpt-5.6-sol high.