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Characterizing transitive G-set actions in terms of orbits

  1. Nov 23, 2011 #1
    1. The problem statement, all variables and given/known data
    A group G acts transitively on a non empty G-set S if, for all s1, s2 in S, there exists an element G in G such that g*s1 = s2. Characterize transitive G-set actions in terms of orbits. Prove your answer


    2. Relevant equations
    Transitive G-set Actions: for all s1, s2 in S, there exists a g in G such that g*s1=s2
    Regular G-set Actions: 1) (gh)s = g(hs)
    2) 1s = s

    Orbit of S = {s' in S such that s' in gs for some g in G}


    3. The attempt at a solution
    Past the definitions, i don't really know anything. The problem is vague and it seems like I'm supposed to write the definition of transitive G-set actions in terms of orbits, but i don't know how to do that nor do i know how i would "prove" that.

    Tips are greatly appreciated :D.
     
  2. jcsd
  3. Nov 23, 2011 #2

    micromass

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    The orbit of an element x is the set

    [tex]\{g\cdot x~\vert~g\in G\}[/tex]

    So the orbit is all the elements where you could send x to. What possible element can you send x to in a transitive action?? What does transitive mean??
     
  4. Nov 23, 2011 #3
    transitive: a(bc) = (ab)c

    and could i send x to itself transitively? say a(a^-1x) = (aa^-1)x?
     
  5. Nov 23, 2011 #4

    micromass

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    No, that's not what transitive is.
     
  6. Nov 23, 2011 #5
    oh shoot. yes thats totally wrong lol...

    if a = b, b = c, then a = c.

    edit: could i be sending x to the whole set of X? could i send it to the whole thing or only one element of X? I'm not sure.
     
    Last edited: Nov 23, 2011
  7. Nov 23, 2011 #6

    micromass

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    No...

    Search in your notes for "transitive action". What is the definition they give. Don't just make things up...
     
  8. Nov 23, 2011 #7
    the only definition i have for transitive action is transitive action on groups and I've posted it above...ahh:/
     
  9. Nov 23, 2011 #8

    micromass

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    You already posted it in the OP... An action is transitive if for each x and y there is a g such that [itex]g\cdot x=y[/itex]. I don't know where the other things come from.

    Now what does transitive mean intuitively?? Can you calculate the orbit right now??
     
  10. Nov 23, 2011 #9
    oh well i thought you were asking for something other than the OP o.o

    intuitively...the transitive action takes one orbit to another orbit?

    the orbit should be {g*x} for x in X
     
  11. Nov 23, 2011 #10

    micromass

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    No, the action always sends an element to the same orbit. By definition.

    Transitivity says that every number can be sent to every other number.

    Now, with this, what is the orbit??
     
  12. Nov 23, 2011 #11

    Deveno

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    if an action is transitive, how can x,y lie in different orbits?
     
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