Finitely generated abelian group

Sajun.org

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. === 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''. === Properties and Classification === Every finitely generated abelian group ''G'' is isomorphic to a direct product of the form :('''Z''')^''n'' × '''Z'''_''m''1 × ... × '''Z'''_''m''''t'' where ''n'' ≥ 0, and the numbers ''m''₁,...,''m''_''t'' are (not necessarily distinct) powers of prime numbers. The values of ''n'', ''m''₁,...,''m''_''t'' are (up to order) uniquely determined by ''G''; in particular, ''G'' is finite if and only if ''n'' = 0 Because of the general fact 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'', we can also write any abelian group ''G'' as a direct product of the form :('''Z''')^''n'' × '''Z'''_''k''1 × ... × '''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''. 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. Every finitely generated abelian group has finite rank equal to the number ''n'' from above. Expressing the theorem in general terms, it says a finitely-generated abelian group is the sum of a free abelian group and a finite abelian group, each of those being unique up to isomorphism. The ''rank'' is an isomorphism invariant. The converse is not true however: there are many abelian groups of finite rank which are not 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.