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Math Help - About Z as cyclic group??

  1. #1
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    About Z as cyclic group??

    Hi, i just wonder how come Z is cyclic group with generator <1> when 1 with addition binary operation can make up Z+ not the negative integers, it's a bit confusing to me i dont understand how come you generate negative numbers with 1??, ofc. you can take -1 and do the trick but as the theory says cyclic group is group such that there exists a E Z (for Z talking now) that whole Z can be generated with that a, so i dont quite see that there exist such unique element in Z that can generate the whole Z in one run... thanks!

    edit: also how can 0 be generated using 1??
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  2. #2
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    Quote Originally Posted by goroner View Post
    Hi, i just wonder how come Z is cyclic group with generator <1> when 1 with addition binary operation can make up Z+ not the negative integers, it's a bit confusing to me i dont understand how come you generate negative numbers with 1??, ofc. you can take -1 and do the trick but as the theory says cyclic group is group such that there exists a E Z (for Z talking now) that whole Z can be generated with that a, so i dont quite see that there exist such unique element in Z that can generate the whole Z in one run... thanks!

    edit: also how can 0 be generated using 1??
    You need to be careful with your definitions!

    You want to show that

    \mathbb{Z}=<1>=\{x=(1)(n), n \in \mathbb{Z} \}

    What I think you are misunderstanding is the meaning of the notation

    (1)(n) what does this mean when n is negative? e.g

    (1)(-k), k > 0 This should be defined in your text!
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    Quote Originally Posted by goroner View Post
    Hi, i just wonder how come Z is cyclic group with generator <1> when 1 with addition binary operation can make up Z+ not the negative integers, it's a bit confusing to me i dont understand how come you generate negative numbers with 1??, ofc. you can take -1 and do the trick but as the theory says cyclic group is group such that there exists a E Z (for Z talking now) that whole Z can be generated with that a, so i dont quite see that there exist such unique element in Z that can generate the whole Z in one run... thanks!

    edit: also how can 0 be generated using 1??

    Additive inverses my friend, additive inverses.
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