A subset X of an abelian group F is said to be linearly independent if n₁x₁ + · · · + n_kx_k = 0 always implies n_i = 0 for all i (where n, ϵ Z and x₁, ... , x_k are distinct elements of X).

(a) X is linearly independent if and only if every nonzero element of the subgroup (X) may be written uniquely in the form n₁x₁ + · · · + n_kX_k (n_i ϵ Z,n_i ≠ 0, x₁, ... , x_k distinct elements of X).
(b) If F is free abelian of finite rank n, it is not true that every linearly independent subset of n elements is a basis

(c) If F is free abelian, it is not true that every linearly independent subset of F may be extended to a basis.
(d) If F is free abelian, it is not true that every generating set of F contains a basis of F. However, if F is also finitely generated by n elements, F has rank m ≤ n.