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Math Help - cyclic group problem??

  1. #1
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    cyclic group problem??

    How to prove that the direct product of group Z is not cyclic?
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  2. #2
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    Can \mathbb{Z}\times\mathbb{Z} be generated by a single element? Suppose it can be generated by an element (a,b) and arrive at contradiction.
    Spoiler:

    if a=0, you'll never get the element (1,0), if a \neq 0, you'll never get the element (a,b+1)
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  3. #3
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    please make more clarity.........
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  4. #4
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    Quote Originally Posted by Taluivren View Post
    Can \mathbb{Z}\times\mathbb{Z} be generated by a single element? Suppose it can be generated by an element (a,b) and arrive at contradiction.
    Spoiler:

    if a=0, you'll never get the element (1,0), if a \neq 0, you'll never get the element (a,b+1)
    please make some more clarity..
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  5. #5
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    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 \mathbb{Z}\times\mathbb{Z} is cyclic or not?

    If \mathbb{Z}\times\mathbb{Z} were cyclic, then there would be an element (a,b) \in \mathbb{Z}\times\mathbb{Z} such that every element (c,d) \in \mathbb{Z}\times\mathbb{Z} can be written as (c,d) = n(a,b) = (na,nb) for some integer n. The proof in the previous post shows that if you choose any (a,b) \in \mathbb{Z}\times\mathbb{Z} then there exists an element from \mathbb{Z}\times\mathbb{Z} such that no such n exists. This contradicts the statement that \mathbb{Z}\times\mathbb{Z} is cyclic.
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  6. #6
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    Quote Originally Posted by Taluivren View Post
    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 \mathbb{Z}\times\mathbb{Z} is cyclic or not?

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