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A point on terminology: a ''free abelian'' group is ''not'' the same as a Free Group that is abelian; in fact the only free groups that are abelian are those of rank 0 (the trivial group) and rank 1 (the infinite cyclic group). If ''F'' is a free abelian group with basis ''B'', then we have the following from ''F'' to ''A'' which extends ''f''. This universal property can also be used to define free abelian groups. For every set ''B'', there exists a free abelian group with basis ''B'', and all such free abelian groups having ''B'' as basis are Isomorphic . One example may be constructed as the abelian group of functions on ''B'', taking Integer values all but finitely many of which are zero. This is the direct sum of copies of Z, one copy for each element of ''B''. Formal sums of elements of a given set ''B'' are nothing but the elements of the free abelian group with basis ''B''. Every . The relationships between different bases can be interesting; for example, the different possibilities for choosing a basis for the free abelian group of rank two is reviewed in the article on the Fundamental Pair Of Periods . Given any abelian group ''A'', there always exists a free abelian group ''F'' and a Surjective group homomorphism from ''F'' to ''A''. This follows from the universal property mentioned above. Importantly, every Subgroup of a free abelian group is free abelian. As a consequence, to every abelian group ''A'' there exists a Short Exact Sequence :0 → ''G'' → ''F'' → ''A'' → 0 with ''F'' and ''G'' being free abelian (which means that ''A'' is isomorphic to the Factor Group ''F''/''G''). This is called a free resolution of ''A''. Furthermore, the free abelian groups are precisely the Projective Objects in the Category Of Abelian Groups . All free abelian groups are but non-zero free abelian groups are never divisible. Free abelian groups are a special case of Free Module s, as abelian groups are nothing but Modules over the Ring Z. It can be surprisingly difficult to determine whether a concretely given group is free abelian. Consider for instance the Baer-Specker Group Z'''N''', the Direct Product of Countably many copies of Z. Reinhold Baer proved in 1937 that this group is ''not'' free abelian; Specker proved in 1950 that every countable subgroup of Z'''N''' is free abelian. SEE ALSO EXTERNAL LINKS |
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