Description
Abelian groups can be examined according to their structure. One of the structures we will investigate is that of finitely generated abelian groups. In particular, the fundamental theorem for finite abelian groups states they can be expressed as a direct sum of primary cyclic groups. These decompositions are not necessarily unique, however they will have the same invariant factors. Similarly, there is a fundamental theorem for finitely generated abelian groups states that these groups can be decomposed into a direct sum of finitely many cyclic subgroups. However, these subgroups can be finite (cyclic p-groups) or infinite (copies of Z, +)). Just as with finite groups, uniqueness is not achieved, but different decompositions will be isomorphic. This means the number of infinite cyclic summands will be the same, as well as the orders of the primary cyclic summands, which define the invariants of the finitely generated abelian group. The property of being finitely generated is closed under sums, subgroups, and quotients. Two important special classes of infinite abelian groups are torsion groups and torsion-free groups. Every torsion group splits into a direct sum of primary groups, and this decomposition is unique. Every quotient group of an abelian group by its torsion subgroup is torsion-free group. So, every abelian group is an extension of a torsion-free group by a torsion group. This extension does not always split, but there are cases in which it does. For example, the torsion subgroup of a finitely generated abelian group is a direct summand. Another example where the extension splits is with divisible abelian groups. Divisible abelian groups can be decomposed into a direct sum of groups isomorphic to the rationals and p∞ groups for various primes p. Divisibility is closed under subgroups, quotients, direct products and sums.
