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.

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