Question: why zxz is not cyclic?
Answer: Proof that Z x Z is not a cyclic group. ... 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.