Thread: 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??

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



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

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

This should be defined in your text!

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??

Additive inverses my friend, additive inverses.