The fundamental theorem of finite Abelian groups states that a finite Abelian group is isomorphic to a direct product of cyclic groups of prime-power order, where the decomposition is unique up to the order in which the factors are written. On the other hand, since the direct product of cyclic groups of relatively prime order is cyclic, there are in fact various ways to express a given finite Abelian group as a direct product of cyclic groups, not necessarily of prime-power order. For example, ≅ , among others. For a given Abelian group of order , with , this Demonstration provides all isomorphism classes. Each class is expressed as the product of cyclic groups of prime-power order guaranteed by the fundamental theorem. The product is also shown in the traditional way, making cyclic factors as large as possible, with the order of each successive factor a divisor of the order of the previous. Note: for a group of prime-power order these two decompositions are the same, so the decomposition is only shown once (see snapshot 4).

Contributed by: Marc Brodie (Wheeling Jesuit University)