Results 1 to 4 of 4

Math Help - Explanation of orbits

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Explanation of orbits

    I have been reading in my book I don't quite understand orbit.

    G(x)=\{gx\in X:g\in G\}

    Can someone explain this and show some easy examples?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Explanation of orbits

    Quote Originally Posted by dwsmith View Post
    I have been reading in my book I don't quite understand orbit.

    G(x)=\{gx\in X:g\in G\}

    Can someone explain this and show some easy examples?
    To create an orbit you just take an element x\in X and apply every group element to it. For example, if G is a finite group and X=\left\{(g_1,\cdots,g_n)\in G^n:g_1\cdots g_n=1\right\} then \mathbb{Z}_n acts on X via cyclic shifting backward of the tuples (i.e. k\cdot(g_1,\cdots,g_n)=(g_{k+1\text{ mod }n},\cdots,g_{k+n\text{ mod }n})). Then, the orbit of (g_1,\cdots,g_n) consists of 0\cdot(g_1,\cdots,g_n)=(g_1,\cdots,g_n), 1\cdot(g_1,\cdots,g_n)=(g_2,g_3,\cdots,g_n,g_1), etc. Make sense?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Explanation of orbits

    Quote Originally Posted by Drexel28 View Post
    To create an orbit you just take an element x\in X and apply every group element to it. For example, if G is a finite group and X=\left\{(g_1,\cdots,g_n)\in G^n:g_1\cdots g_n=1\right\} then \mathbb{Z}_n acts on X via cyclic shifting backward of the tuples (i.e. k\cdot(g_1,\cdots,g_n)=(g_{k+1\text{ mod }n},\cdots,g_{k+n\text{ mod }n})). Then, the orbit of (g_1,\cdots,g_n) consists of 0\cdot(g_1,\cdots,g_n)=(g_1,\cdots,g_n), 1\cdot(g_1,\cdots,g_n)=(g_2,g_3,\cdots,g_n,g_1), etc. Make sense?
    Nope
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,371
    Thanks
    740

    Re: Explanation of orbits

    the motivating example of a group acting on a set is G = Sn, where X is the set {1,2,...,n}. this action is transitive, which means for any element x of X, G(x) = X.

    so let's consider a subgroup of G, say H = <(1 2 3)>. what is the orbit H(1)? well, if h = (1 2 3), h(1) = 2, so 2 is in the orbit of 1. similarly h^2 = (1 3 2), so h^2(1) = 3,

    so 3 is in the orbit of 1. it should be pretty clear that if x is not 1,2 or 3, then h(x) = x. in particular none of {4,....,n} are in the orbit of 1. obviously the identity map sends 1-->1,

    so 1 is in the orbit of 1. so the orbit of 1 under H, H(1) = {1,2,3}.

    we can also consider an action of G on itself, by defining for g in G and x in X (=G), g(x) = gx, the product in G. similarly, we can also let a subgroup H act on G the same way:

    define h(g) = hg (the product in G). let's take G = Z10, H = Z5, and look at H(4).

    the elements of H are: {0,5}. if h = 0, h(4) = 0+4 = 4, so 4 is in the orbit of 4. if h = 5, then h(4) = 5+4 = 9, so H(4) = {4,9}. note that in this case, H(n) is the coset H+n.

    note also that H(9) = {0+9,5+9} = {9,4} = H(4), perhaps this makes clear the term "orbit"...the action of H on G partitions G into disjoint sets, and on each of these orbits,

    H sends an element of that orbit to another element of that orbit.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Orbits and Groups
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 3rd 2010, 12:33 AM
  2. help w/ Orbits and Stabilizers
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 4th 2009, 07:53 PM
  3. ODE's and orbits
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: December 6th 2008, 02:05 PM
  4. Periodic Orbits
    Posted in the Calculus Forum
    Replies: 0
    Last Post: November 29th 2006, 10:43 AM
  5. group orbits
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: July 31st 2005, 06:20 PM

Search Tags


/mathhelpforum @mathhelpforum