All cyclic groups are abelian groups. It is not true that every abelian group is cyclic.
Proof. Take the cyclic group G = \langle g \rangle. Take the elements a,b \in G. We need to show that ab = ba. Since G is cyclic, we have that a = g^i and b = g^j for some i,j \in \mathbb{Z}. This implies that ab = g^ig^j = g^{i + j} = g^{j + i} = g^jg^i = ba.
This proves that cyclic groups are abelian groups.
