Thread: cyclic group problem??

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

Can be generated by a single element? Suppose it can be generated by an element and arrive at contradiction.

Spoiler:

if , you'll never get the element , if , you'll never get the element

please make more clarity.........

Originally Posted by Taluivren

Can be generated by a single element? Suppose it can be generated by an element and arrive at contradiction.

Spoiler:

if , you'll never get the element , if , you'll never get the element

please make some more clarity..

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.

Originally Posted by Taluivren

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.

thanks.........