Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 2G / Solution

The Fundamental theorem of finitely generated abelian groups expresses every finite abelian group of this order as a product of cyclic groups of prime-power order. Their exponents correspond to the partitions of an integer $4$:These are pairwise nonisomorphic: the largest order of a group element distinguishes all except the two groups with largest element order $p^2$, and those have respectively $p^2$ and $p^3$ elements annihilated by multiplication by $p$. The classification exhausts the possibilities, so there are five isomorphism classes.