1883. PRODUCTS OF GROUPScombination of finitely many elements ofB. SinceSis finite, it is con-tained in the subgroup geneated by a finite subsetB0ofB.But thenGDZSZB0.SoB0generatesG.It follows from the previouslemma thatB0DB.nProposition 3.5.5.Any two bases of a finitely generated free abelian grouphave the same cardinality.Proof.LetGbe a finitely generated free abelian group. By the previouslemma, any basis ofGis finite. IfGhas a basis withnelements, thenGŠZn, by Proposition3.5.2. SinceZnandZmare nonisomorphic ifm¤n,Gcannot have bases of different cardinalities.nDefinition 3.5.6.Therankof a finitely generated free abelian group is thecardinality of any basis.Proposition 3.5.7.Every subgroup ofZncan be generated by no morethannelements.Proof.The proof goes by induction onn. We know that every subgroup ofZis cyclic (Proposition2.2.21), so this takes care of the base casenD1.Suppose thatn > 1and that the assertion holds for subgroups of
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