1. ## 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??

2. Originally Posted by goroner
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!

3. Originally Posted by goroner
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??