finitely generated abelian group

Among abelian groups, the finitely generated ones are completely understood — there is a clean classification leaving no surprises. Each such group splits into a “free” part counting independent directions of infinite extent, plus a finite “torsion” part recording the cyclic loops, and the whole group is just a direct sum of cyclic pieces.

Formally, an abelian group A is finitely generated if there are finitely many elements g_1, ..., g_n with every element of A an integer combination of them. The structure theorem then says A is isomorphic to Z^r ⊕ Z/d_1Z ⊕ ... ⊕ Z/d_kZ where r ≥ 0 is the rank (the free part) and d_1 | d_2 | ... | d_k are the invariant factors. Equivalently, regrouping by primes, A is a sum of Z^r with cyclic groups of prime-power order (the elementary divisor form). The data (r; d_1, ..., d_k) is a complete isomorphism invariant.

This is the abelian special case of the structure theorem for finitely generated modules over a principal ideal domain, with the PID being Z. It is the engine behind, for example, the classification of finitely generated modules over Euclidean domains and the rational/Jordan canonical forms of a matrix (taking the PID to be a polynomial ring). The hypothesis “finitely generated” is essential — infinitely generated abelian groups such as the rationals Q under addition are not classified by any such finite list of invariants.