# Thread: Cyclic Group having only one generator

1. ## Cyclic Group having only one generator

Good day,.,.is it possible for a cyclic group to have only one generator???

2. ## Re: Cyclic Group having only one generator

It's not just possible. That's the definition of a cyclic group. Cyclic groups are groups generated by a single element.

3. ## Re: Cyclic Group having only one generator

,.,thnx a lot sir,.,ammm,..what does it really mean when u said "generated by a single element"??isn't it the same with "one generator"??thnx a lot sir,.,sorry if im to confused with this topic,.,

4. ## Re: Cyclic Group having only one generator

So far as I know, the phrase "one generator" and "generated by a single element" mean the same thing.

[EDIT]: See below for a correction of the understanding here.

I'm not an algebraist, but that's my understanding. The wiki page to which I linked before has a precise definition of the term. It says, "In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g (called a 'generator' of the group) such that, when written multiplicatively, every element of the group is a power of g (a multiple of g when the notation is additive)."

5. ## Re: Cyclic Group having only one generator

,.,yupzz,.,.it really answer my question,.,.therefore,.can i say that any example of cyclic group that i give has only one generator??

6. ## Re: Cyclic Group having only one generator

Oops. I have to be careful here. A cyclic group can be generated by only one generator. However, it may not be the case that every element in a cyclic group generates the whole group. In addition, more than one element in the group may generate the group. In $\displaystyle \mathbb{Z}_{5},$ for example, every element except the identity generates the group.

7. ## Re: Cyclic Group having only one generator

,.ohhhh,..,so does that mean that Z5 contains four generators???

8. ## Re: Cyclic Group having only one generator

Originally Posted by aldrincabrera
,.ohhhh,..,so does that mean that Z5 contains four generators???
Correct, since the identity can only generate the trivial subgroup consisting of itself.

9. ## Re: Cyclic Group having only one generator

,.,.thnx a lot sir,.,now i understand,.,.one last question sir,.,.i find it easy now to find examples of cyclic groups with more than one generator such as Z7 and Z6,.,.but im having a hard tym looking for a cyclic group with one generator,.,can u give me an example??thnk u so much for ur help sir,.,

10. ## Re: Cyclic Group having only one generator

Originally Posted by aldrincabrera
,.,.thnx a lot sir,.,now i understand,.,.one last question sir,.,.i find it easy now to find examples of cyclic groups with more than one generator such as Z7 and Z6,.,.but im having a hard tym looking for a cyclic group with one generator,.,can u give me an example??thnk u so much for ur help sir,.,
How about $\displaystyle \mathbb{Z}_{2}?$ That might be the only cyclic group with only one generator.

11. ## Re: Cyclic Group having only one generator

,.,.ohhh,..yeah ,.,having {e,1} as elements,.,surely i can only have the element 1 as generator,.,.thank u so much sir,.,

12. ## Re: Cyclic Group having only one generator

I would write that group additively as {0,1} using addition modulo 2, or multiplicatively as {-1,1} using regular multiplication. I think {e,1} might be a bit confusing unless it's understood that you're talking about a multiplicative group, and e = 0.

You're very welcome for whatever help I could provide.

13. ## Re: Cyclic Group having only one generator

,.,.thnx sir,.,i'll just be using addition modulo 2,.,

14. ## Re: Cyclic Group having only one generator

Originally Posted by aldrincabrera
,.,.thnx sir,.,i'll just be using addition modulo 2,.,
You're welcome. Have a good one!

15. ## Re: Cyclic Group having only one generator

the number of generators of a cyclic group of order $\displaystyle n$ is $\displaystyle \varphi(n)$ and the number of generators of an infinite cyclic group is $\displaystyle 2$. to see this, suppose $\displaystyle G = \langle x \rangle$ is a cyclic group of order $\displaystyle n$. then we know that the order of an element of $\displaystyle G$, say $\displaystyle x^k$, is $\displaystyle \frac{n}{\gcd(k,n)}$. so $\displaystyle x^k$ is a generator of $\displaystyle G$ if and only if $\displaystyle \gcd(k,n)=1$ and this proves what i said.
if you want your group to have one generator only, it means $\displaystyle \varphi(n)=1$ which has two solutions: $\displaystyle n = 1,2.$

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# show that a cyclic group with only one generator can have atmost 2 elements

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