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Math Help - Subgrounds of Symmetric Groups

  1. #1
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    Subgrounds of Symmetric Groups

    I'm trying to find all of the subgroups of the symmetric S_{3}. But am not sure how to generate these groups as all of the properties of groups must be satisfied.

    I am thinking it is something along the lines of:
    {(123)->(123)} (no change)
    {(123)->(123), (312), (231)}
    {(123)->(123),(132), (213), (321)}

    Is this the right track? Or am out out in left field?
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  2. #2
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    Quote Originally Posted by apsis View Post
    I'm trying to find all of the subgroups of the symmetric S_{3}. But am not sure how to generate these groups as all of the properties of groups must be satisfied.

    I am thinking it is something along the lines of:
    {(123)->(123)} (no change)
    {(123)->(123), (312), (231)}
    {(123)->(123),(132), (213), (321)}

    Is this the right track? Or am out out in left field?
    The subgroups are \{ e \}, \left< (12) \right>, \left< (13)\right>, \left< (23) \right>, \left< (123) \right>,S_3.
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