We prove that the only abelian groups which appear as fundamental groups of closed, connected, orientable $3$-manifolds are
$$\Bbb Z=\pi_1(\Bbb S^1\times \Bbb S^2),\ \Bbb Z_n=\pi_1\big(L(1,n)\big),\text{ and }\Bbb Z^3=\pi_1(\Bbb S^1\times \Bbb S^1\times \Bbb S^1).$$
$3$-manifolds, Heegaard splittings, decomposition of $3$-manifolds