Hi.
if G = <a> then Every subgroup of G is Cyclic so generated by one element (finitely generated).
if the hypothesis is true for abelian groups that generated by less than n elements (n>=2) and G = <a1,...,an> then let H = <a1,...,an-1> and let A < G.
Now let R = {dn | d1*a1+...+dn*an is an...