Mark each of the following true or false. 

___ a. Every free abelian group is torsion free. 

___ b. Every finitely generated torsion-free abelian group is a free abelian group. 

___ c. There exists a free abelian group of every positive integer rank. 

___ d. A finitely generated abelian group is free abelian if its Betti number equals the number of elements in some generating set. 

___ e. If X generates a free abelian group G and X ⊆ Y ⊆ G, then Y generates G. 

___ f. If X is a basis for a free abelian group G and X ⊆ Y ⊆ G, then Y is a basis for G. 

___ g. Every nonzero free abelian group has an infinite number of bases. 

___ h. Every free abelian group of rank at least 2 has an infinite number of bases. 

___ i. If K is a nonzero subgroup of a finitely generated free abelian group, then K is free abelian. 

___ j. If K is a nonzero subgroup of a finitely generated free abelian group, then G/K is free abelian.