Proof that Z x Z is not a cyclic group

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Proof that Z x Z is not a cyclic group. We use a proof by contradiction.

It follows that the direct product of ANY two infinite cyclic groups is not cyclic. To see this let ~ denote isomorphism and note that if G and H are infinite cyclic groups, then G ~ Z and H ~ Z. It follows that G x H ~ Z x Z which is not cyclic, hence G x H is not cyclic either.

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