How to prove that the direct product of group Z is not cyclic?

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- Sep 8th 2009, 11:19 PMMathventurecyclic group problem??
How to prove that the direct product of group Z is not cyclic?

- Sep 9th 2009, 12:15 AMTaluivren
Can $\displaystyle \mathbb{Z}\times\mathbb{Z}$ be generated by a single element? Suppose it can be generated by an element $\displaystyle (a,b)$ and arrive at contradiction.

__Spoiler__: - Sep 9th 2009, 04:21 AMMathventure
please make more clarity.........

- Sep 9th 2009, 04:25 AMMathventure
- Sep 9th 2009, 04:38 AMTaluivren
Hi, what exactly is not clear?

by definition, a cyclic group is a group that can be generated by a single element. As I understand, you ask whether $\displaystyle \mathbb{Z}\times\mathbb{Z}$ is cyclic or not?

If $\displaystyle \mathbb{Z}\times\mathbb{Z}$ were cyclic, then there would be an element $\displaystyle (a,b) \in \mathbb{Z}\times\mathbb{Z}$ such that every element $\displaystyle (c,d) \in \mathbb{Z}\times\mathbb{Z}$ can be written as $\displaystyle (c,d) = n(a,b) = (na,nb)$ for some integer $\displaystyle n$. The proof in the previous post shows that if you choose any $\displaystyle (a,b) \in \mathbb{Z}\times\mathbb{Z}$ then there exists an element from $\displaystyle \mathbb{Z}\times\mathbb{Z}$ such that no such $\displaystyle n$ exists. This contradicts the statement that $\displaystyle \mathbb{Z}\times\mathbb{Z}$ is cyclic. - Sep 10th 2009, 05:19 AMMathventure