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.