The theorem of free abelian group says,
"If F is a free abelian group of finite rank n and G is a nonzero subgroup of F, then there exists a basis of F, an integer and positive integers such that and G is free abelian with basis ."
My question is
If I choose a basis for G as and ,respectively, G satisfies the above condition and I see that G is a subgroup of F.
If I choose a basis for G as , G does not satisfies the above theorem. But G looks still a subgroup of F to me.
The members of G might be
Please correct me if I am wrong on the concept of the above theorem.
Thanks in advance.