How we can prove additive group of rational numbers (Q,+)is not cyclic, but every finitely generated subgroup of (Q,+)is cyclic.
This question obviously belongs in Linear and Abstract Algebra and not here.
Assume Q = <a/b> and check the prime factorization of b. Now show that
any multiple m*(a/b) of a/b cannot have different primes in the denominator from the ones that appear in the decomposition of b and thus...
About every fin. gen. sbgp.: check first for a 2-generator group (pretty easy) and then generalize by induction