Results 1 to 6 of 6

Math Help - Orbits and Groups

  1. #1
    Member
    Joined
    Oct 2007
    Posts
    209

    Orbits and Groups

     \chi = {1,2,3,4,5,6,7}
    [tex] g=(1 2) (3 4 5 6)
    Let G= [tex] {1, g, g^2, g^3}

    Compute  g^2 and g^3
    Computer both stabilizer and orbit of 1,3,7 in G.

    Note:
    |stabilizer (x)||orbit (x)|=|G|
    Stabilizers is the Action
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by Linnus View Post
     \chi = {1,2,3,4,5,6,7}
    [tex] g=(1 2) (3 4 5 6)
    Let G= [tex] {1, g, g^2, g^3}

    Compute  g^2 and g^3
    Computer both stabilizer and orbit of 1,3,7 in G.

    Note:
    |stabilizer (x)||orbit (x)|=|G|
    Stabilizers is the Action
    The orbit of an element in the set the group is acting upon, S, is everywhere in S it is sent to by an element of the group. So, for example, there is no way of sending 1 to anything other than 2. Similary, g sends 1 to 2.

    The stabiliser of an element is the set of all group elements which "stabilise" the element; group elements which don't send the element anywhere.

    So, for example, 1g=2 = 1g^3 but 1g^2 = 1 = 1e. Therefore, g^2 stabilises 1, as does the group identity. Therefore, the orbit of the element 1 is {1, 2} and its stabiliser is the subgroup \{e, g^2\}. Note that 2.2=4=|G|.

    Does that make sense?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2007
    Posts
    209
    Thanks for the help. I understand that part, but I don't know how to compute the orbit and the stabilizer of 3. Any help is appreciated!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Mar 2010
    Posts
    28
    [quote=Linnus;505474]  \chi = {1,2,3,4,5,6,7}
    [tex] g=(1 2) (3 4 5 6)
    Let G= [tex] {1, g, g^2, g^3}

    Compute  g^2 and g^3
    g^2 = (1 2) (3 4 5 6)(1 2) (3 4 5 6) = (3 5)(4 6)
    g^3 = (3 5)(4 6)(1 2) (3 4 5 6) = (3 6 5 4)

    stab(1) = {g^2,g^3}, orb(1) = 2
    stab(3) = {1}, orb(3) = 4
    stab(7) = {1,g,g^2,g^3}, orb(7) = 1

    i am not sure, but thats what i would do
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2010
    Posts
    28
    Quote Originally Posted by Linnus View Post
    Thanks for the help. I understand that part, but I don't know how to compute the orbit and the stabilizer of 3. Any help is appreciated!

    stabilizer of x = consist of those elements of G that send x itself,
    which is neutral element in this case (for 3)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by Linnus View Post
    Thanks for the help. I understand that part, but I don't know how to compute the orbit and the stabilizer of 3. Any help is appreciated!
    I you understand my post, then surely you can apply it to working out the orbit and stabiliser of 3?

    What is it, specifically, that you do not understand?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Dependency between orbits
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 20th 2011, 10:20 AM
  2. Explanation of orbits
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 2nd 2011, 11:31 PM
  3. help w/ Orbits and Stabilizers
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 4th 2009, 08:53 PM
  4. ODE's and orbits
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: December 6th 2008, 03:05 PM
  5. group orbits
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: July 31st 2005, 07:20 PM

Search Tags


/mathhelpforum @mathhelpforum