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Math Help - simple question regarding orbits

  1. #1
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    simple question regarding orbits

    This topic is giving me a hard time, I was thinking an example might help clear thing up.

    Lets say I was given a group S_3 which consists of \{1, (1,2), (1,3), (2,3), (1,2,3), (1,3,2)\}

    and a set X consisting of \{2,3,7,9\}

    and I was asked to find the respective orbits. Then would it simply be:

    1\cdot 2 = 2 where 1 is a stabalizer
    (1,2)(2) = (1,2)
    (1,3)(2) = (1,3) stabalizes 2
    (2,3)(2) = (2,3)
    (1,2,3)(2) = (1,2,3)
    (1,3,2)(2) = (1,3,2)

    so the orbit of 2 is \{1,2,3\} with order 3. And if I do the same procedure for 3 I would get the same elements with the same order so O_2 = O_3 = \{1,2,3\}

    Regarding 7 and 9, I'm thinking that all elements of S_3 stabalize 7 and 9, since they don't move, both with order 1 and O_7 = \{7\} and O_9 = \{9\}

    is this correct?

    And how would S_3 partition X?
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  2. #2
    Senior Member Shanks's Avatar
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    consider the action of S_3 on X.
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