Finitely Generated Abelian Groups

An element is torsion when some positive integer multiple of it is zero. In an abelian group, torsion elements form a subgroup: if integers m and n annihilate x and y, then mn annihilates both their sum and their difference. Denote this subgroup by T. The group of integers has no nonzero torsion; a finite cyclic group consists entirely of torsion. A mixed example is an integer coordinate paired with a residue modulo twelve. Moving in the integer coordinate never returns to zero unless the move itself is zero.

The structure theorem says a finitely generated abelian group is isomorphic to a finite direct sum of copies of the integers and finite cyclic groups. Its free rank r counts the integer coordinates. Its torsion subgroup is the finite collection of modular coordinates, and quotienting by torsion leaves a group isomorphic to Z^r. The rank and the isomorphism type of T are intrinsic. A particular choice of free generators or a splitting inside the original group need not be canonical. Finite generation matters: the additive rationals are torsion-free but are not a finite-rank free abelian group.

Invariant factors arrange cyclic orders in a divisibility chain. Elementary divisors instead split those orders into prime powers. The bridge is the Chinese remainder theorem: when a and b are coprime, reduction modulo a and modulo b identifies a residue modulo ab with a pair of residues. Coprimality is essential. Two independent residues modulo two cannot produce an element of order four, so their direct sum is not cyclic of order four. Cardinality alone misses that distinction.

For example, cyclic factors of orders six and twelve already form an invariant-factor description because six divides twelve. Splitting each factor gives prime-power factors of orders two, three, four, and three. To reconstruct invariant factors, sort the powers for each prime increasingly, pad shorter lists with ones on the left, and multiply entries in matching positions. This yields six and twelve again. Prime-power factors are unique as a multiset; arbitrary unsorted cyclic factorizations are not the classification data. The largest invariant factor is the exponent of the finite part: the smallest positive integer annihilating every torsion element.

Choose n generators and map integer coordinate vectors onto the group. Relations form the kernel, a subgroup of Z^n; the group is the quotient by that kernel. Integer row and column operations corresponding to invertible integer changes of generators and relations preserve the quotient up to isomorphism. Euclidean division reduces a relation matrix to Smith normal form: a diagonal matrix whose nonzero positive entries divide the next ones. These diagonal entries give the cyclic factors, while directions with no diagonal relation remain free. This explains the theorem's existence mechanism without pretending that rational Gaussian elimination preserves an integer lattice.

Uniqueness can be tested through information preserved by isomorphisms. For a finite p-primary part, count elements annihilated by p, then by its successive powers. In a cyclic factor of order p^a, the number annihilated by p^k is p raised to the smaller of a and k. Taking base-p logarithms of the product counts gives successive sums; their differences recover how many cyclic factors have exponent at least k. Thus the elementary-divisor exponents are recoverable from the abstract group itself. This is stronger than comparing total order and provides a practical check on proposed classifications.