FINITELY GENERATED ABELIAN GROUP

In abstract algebra, an abelian group (''G'',+) is called 'finitely generated' if there exist finitely many elements ''x''₁,...,''x''_''s'' in ''G'' such that every ''x'' in ''G'' can be written in the form
:''x'' = ''n''₁''x''₁ + ''n''₂''x''₂ + ... + ''n''_''s''''x''_''s''
with integers ''n''₁,...,''n''_''s''. In this case, we say that the set {''x''₁,...,''x''_''s''} is a ''generating set'' of ''G'' or that ''x''₁,...,''x''_''s'' ''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.

Contents

Examples

Classification

Primary decomposition

Invariant factor decomposition

Equivalence

Corollaries

Non-finitely generated abelian groups

See also

Examples

★ the integers ('Z',+) are a finitely generated abelian group

★ the integers modulo ''n'' 'Z'_''n'' are a finitely generated abelian group

★ any direct sum of finitely many finitely generated abelian groups is again finitely generated abelian
There are no other examples. The group ('Q',+) of rational numbers is not finitely generated: if ''x''₁,...,''x''_''s'' are rational numbers, pick a natural number ''w'' coprime to all the denominators; then 1/''w'' cannot be generated by ''x''₁,...,''x''_''s''.

Classification

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 PIDs):

Primary decomposition

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 such group is isomorphic to one of the form
:$mathbb\{Z\}^n\; oplus\; mathbb\{Z\}\_\{q\_1\}\; oplus\; cdots\; oplus\; mathbb\{Z\}\_\{q\_t\}$
where ''n'' ≥ 0, and the numbers ''q''₁,...,''q''_''t'' are (not necessarily distinct) powers of prime numbers. In particular, ''G'' is finite if and only if ''n'' = 0. The values of ''n'', ''q''₁,...,''q''_''t'' are (up to order) uniquely determined by ''G''.

Invariant factor decomposition

We can also write any abelian group ''G'' as a direct product of the form
:$mathbb\{Z\}^n\; oplus\; mathbb\{Z\}\_\{k\_1\}\; oplus\; cdots\; oplus\; mathbb\{Z\}\_\{k\_u\}$
where ''k''₁ divides ''k''₂, which divides ''k''₃ and so on up to ''k''_''u''. Again, the numbers ''n'' and ''k''₁,...,''k''_''u'' are uniquely determined by ''G'' (here with a unique order), and are called invariant factors.

Equivalence

These statements are equivalent because of the Chinese remainder theorem, which here states that 'Z'_''m'' is isomorphic to the direct product of 'Z'_''j'' and 'Z'_''k'' if and only if ''j'' and ''k'' are coprime and ''m'' = ''jk''.

Corollaries

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: 'Q' 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 'Q' is one example, and the rank-0 group given by a direct sum of countably many copies of 'Z'₂ is another one.

See also

★ The Jordan-Hölder theorem is a non-abelian generalization