Let G be a finitely generated abelian group with identity 0. A finite set {b₁, · · · , b_n}, where b_i ∈ G, is a basis for G if {b₁, · · · , b_n} generates G and ∑ⁿ_i=1 =1 m_ib_i = 0 if and only if each m_ib_i = 0, where m_i ∈ Z. 

a. Show that {2, 3} is not a basis for Z₄ . Find a basis for Z₄ . 

b. Show that both {1} and {2, 3} are bases for Z₆. (This shows that for a finitely generated abelian group G with torsion, the number of elements in a basis may vary; that is, it need not be an invariant of the group G.) 

c. Is a basis for a free abelian group as we defined it in Section 38 a basis in the sense in which it is used in this exercise? 

d. Show that every finite abelian group has a basis {b₁, · · · , b_n}, where the order of b_i divides the order of b_{i + 1} .