Prove or disprove if G is an abelian and H is a normal subgroup of G,then H must be a cyclic

Question

$G$ is an abelian group,so every subgroup of an abelian group is normal and: $$o(H)|o(G)$$ $G$ is an abelian group:$$\forall a,b:ab=ba$$ $H$ is a normal subgroup:$$\forall g \in G,\forall h \in H: ghg^{-1} \in H$$ $$and$$ $$gH=Hg$$ I need to show there is a generator in H,how? $$$$Thanks.

Answer 1

Every group $G$ is a normal subgroup of itself, so this would imply that every abelian group is cyclic.

$\mathbb R$ with addition is a counterexample.

Answer 2

Any subgroup of an abelian group is normal. So to find a counterexample, you could take any non-cyclic subgroup of any abelian group.

For instance, $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$ is an abelian group, and $\mathbb{Z} \times \mathbb{Z} \times \{0\}$ is an acyclic normal subgroup.