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Thread: 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 $\displaystyle \mathbb{Z}\times\mathbb{Z}$ be generated by a single element? Suppose it can be generated by an element $\displaystyle (a,b)$ and arrive at contradiction.
    Spoiler:

    if $\displaystyle a=0$, you'll never get the element $\displaystyle (1,0)$, if $\displaystyle a \neq 0$, you'll never get the element $\displaystyle (a,b+1)$
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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 $\displaystyle \mathbb{Z}\times\mathbb{Z}$ be generated by a single element? Suppose it can be generated by an element $\displaystyle (a,b)$ and arrive at contradiction.
    Spoiler:

    if $\displaystyle a=0$, you'll never get the element $\displaystyle (1,0)$, if $\displaystyle a \neq 0$, you'll never get the element $\displaystyle (a,b+1)$
    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 $\displaystyle \mathbb{Z}\times\mathbb{Z}$ 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.
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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 $\displaystyle \mathbb{Z}\times\mathbb{Z}$ 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.
    thanks.........
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