a proving question?

Question

Let G be a group of order ab where a and b are distinct primes. Prove that every proper subgroup of G is cyclic.

Answer 1

Easy, use Sylow's or Lagrange's theorem. The only possible proper, nontrivial subgroups have order a or b where those are prime. Subgroups of prime order are cyclic. Also the trivial subgroup is cyclic (of order 1). Hence proved.