Results 1 to 15 of 15

Math Help - Cyclic Group having only one generator

  1. #1
    Member aldrincabrera's Avatar
    Joined
    Jun 2011
    From
    Dumaguete City
    Posts
    90

    Smile Cyclic Group having only one generator

    Good day,.,.is it possible for a cyclic group to have only one generator???
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member aldrincabrera's Avatar
    Joined
    Jun 2011
    From
    Dumaguete City
    Posts
    90

    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,.,
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    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)."

    Does that answer your question?
    Last edited by Ackbeet; August 11th 2011 at 05:44 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member aldrincabrera's Avatar
    Joined
    Jun 2011
    From
    Dumaguete City
    Posts
    90

    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??
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    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 \mathbb{Z}_{5}, for example, every element except the identity generates the group.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member aldrincabrera's Avatar
    Joined
    Jun 2011
    From
    Dumaguete City
    Posts
    90

    Re: Cyclic Group having only one generator

    ,.ohhhh,..,so does that mean that Z5 contains four generators???
    Follow Math Help Forum on Facebook and Google+

  8. #8
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    Re: Cyclic Group having only one generator

    Quote Originally Posted by aldrincabrera View Post
    ,.ohhhh,..,so does that mean that Z5 contains four generators???
    Correct, since the identity can only generate the trivial subgroup consisting of itself.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member aldrincabrera's Avatar
    Joined
    Jun 2011
    From
    Dumaguete City
    Posts
    90

    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,.,
    Follow Math Help Forum on Facebook and Google+

  10. #10
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    Re: Cyclic Group having only one generator

    Quote Originally Posted by aldrincabrera View Post
    ,.,.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 \mathbb{Z}_{2}? That might be the only cyclic group with only one generator.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member aldrincabrera's Avatar
    Joined
    Jun 2011
    From
    Dumaguete City
    Posts
    90

    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,.,
    Follow Math Help Forum on Facebook and Google+

  12. #12
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    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.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Member aldrincabrera's Avatar
    Joined
    Jun 2011
    From
    Dumaguete City
    Posts
    90

    Re: Cyclic Group having only one generator

    ,.,.thnx sir,.,i'll just be using addition modulo 2,.,
    Follow Math Help Forum on Facebook and Google+

  14. #14
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2

    Re: Cyclic Group having only one generator

    Quote Originally Posted by aldrincabrera View Post
    ,.,.thnx sir,.,i'll just be using addition modulo 2,.,
    You're welcome. Have a good one!
    Follow Math Help Forum on Facebook and Google+

  15. #15
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Re: Cyclic Group having only one generator

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

Similar Math Help Forum Discussions

  1. Definition of a generator of a cyclic group?
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 10th 2011, 07:18 PM
  2. Finding a generator of a cyclic group
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 16th 2011, 07:08 PM
  3. Finding a Generator of a Cyclic Group
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: March 9th 2011, 06:40 PM
  4. [SOLVED] Cyclic groups of order greater then 2 have more then one generator
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 12th 2010, 07:12 PM
  5. Prove cyclic subroups => cyclic group
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: October 11th 2009, 07:36 PM

Search Tags


/mathhelpforum @mathhelpforum