Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 1E (Groups, Rings and Modules)

The Smith normal form of an integer $m\times n$ matrix $A$ is a diagonal matrixfor whichwith $U\in G{L}_m(\mathbb{Z})$ and $V\in G{L}_n(\mathbb{Z})$. It exists and is unique.

The structure theorem for finitely generated modules over a principal ideal domain, specialized to $\mathbb{Z}$, says that every finitely generated abelian group has a unique invariant-factor decompositionEquivalently, its finite part is a direct sum of cyclic groups of prime-power order.

Integer row and column operations giveThus the Smith normal form isThe two relation vectors are the columns of the displayed matrix, so its cokernel is the presented module. Consequently

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