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

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

- September 9th 2009, 12:15 AMTaluivren
Can be generated by a single element? Suppose it can be generated by an element and arrive at contradiction.

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

- September 9th 2009, 04:25 AMMathventure
- September 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 is cyclic or not?

If were cyclic, then there would be an element such that every element can be written as for some integer . The proof in the previous post shows that if you choose any then there exists an element from such that no such exists. This contradicts the statement that is cyclic. - September 10th 2009, 05:19 AMMathventure