Finitely generated abelian group

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In abstract algebra, an abelian group (G, +) is called finitely generated if there exist finitely many elements x1, ..., xs in G such that every x in G can be written in the form

x = n1x1 + n2x2 + ... + nsxs

with integers n1, ..., ns. In this case, we say that the set {x1, ..., xs} is a generating set of G or that x1, ..., xs generate G.

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:[1] 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 groups of real numbers under addition and non-zero real numbers under multiplication are also not finitely generated.[1][2]


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):

Primary decomposition[edit]

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[edit]

We can also write any finitely generated abelian group G as a direct sum of the form

where k1 divides k2, which divides k3 and so on up to ku. Again, the rank n and the invariant factors k1, ..., ku are uniquely determined by G (here with a unique order).


These statements are equivalent because of the Chinese remainder theorem, which here states that if and only if j and k are coprime and m = jk.


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[edit]

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.

See also[edit]


  1. ^ a b Silverman & Tate (1992), p. 102
  2. ^ de la Harpe (2000), p. 46