I haven't done this stuff in a while but since no one is helping I'll give it a shot.
Assume not. Then by Fundamental theorem of finitely generated abelian groups, H is isomorphic to Z_a x Z_b. Since, by assumption, H is not cyclic, it follows that Z_a x Z_b is not cyclic. This implies that a and b are not relatively prime. Which implies that there exists some prime p that divides both a and b. Perhaps apply Cauchy's theorem here.