Finitely-generated abelian group
- x = n1x1 + n2x2 + ... + nsxs
Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.
- The integers are a finitely generated abelian group.
- The integers modulo , are a finitely generated abelian group.
- Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group.
- Every lattice forms a finitely-generated free abelian group.
There are no other examples (up to isomorphism). In particular, the group of rational numbers is not finitely generated: if are rational numbers, pick a natural number coprime to all the denominators; then cannot be generated by . The group of non-zero rational numbers is also not finitely generated.
The fundamental theorem of finitely generated abelian groups (which is a special case of the structure theorem for finitely generated modules over a principal ideal domain) can be stated two ways (analogously with principal ideal domains):
The primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups. A primary cyclic group is one whose order is a power of a prime. That is, every finitely generated abelian group is isomorphic to a group of the form
where the rank n ≥ 0, and the numbers q1,...,qt are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q1,...,qt are (up to rearranging the indices) uniquely determined by G.
Invariant factor decomposition
We can also write any finitely generated abelian group G as a direct sum of the form
Stated differently the fundamental theorem says that a finitely-generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G. The rank of G is defined as the rank of the torsion-free part of G; this is just the number n in the above formulas.
A corollary to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian. The finitely generated condition is essential here: is torsion-free but not free abelian.
Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the group homomorphisms, form an abelian category which is a Serre subcategory of the category of abelian groups.
Non-finitely generated abelian groups
Note that not every abelian group of finite rank is finitely generated; the rank 1 group is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of is another one.
- The Jordan–Hölder theorem is a non-abelian generalization
- Silverman & Tate (1992),
- La Harpe (2000),
- Silverman, Joseph H.; Tate, John Torrence (1992). Rational points on elliptic curves. Undergraduate texts in mathematics. Springer. ISBN 978-0-387-97825-3.
- La Harpe, Pierre de (2000). Topics in geometric group theory. Chicago lectures in mathematics. University of Chicago Press. ISBN 978-0-226-31721-2.