Modern Algebra: Finitely generated abelian groups

Exercise Il. Let (a,,.. , an) be a basis of a free abelian group A and B a subgroup of A generated by bis.., th, where 6. = Zda. Prove that C= 1 (a) A/ Bis finite iff det ( 2:; ) + 0 ( 8) if det (a; ) = mto , then | A/ B| = 1ml, ( c ) ( lis.. , bu ) is a basis of A iff def (dij ) = # 1