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Math Help - The group S6

  1. #1
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    The group S6

    I wanted to ask about 3sylowsubgroup (of the group S6)
    I have an example of such a group,
    But I want to know
    what another group is isomorphous? (otherwise known group which does not belong to S6)
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rtrt1 View Post
    I wanted to ask about 3sylowsubgroup (of the group S6)
    I have an example of such a group,
    But I want to know
    what another group is isomorphous? (otherwise known group which does not belong to S6)
    Sorry, what are you asking? Are you wanting to know what the Sylow-3 subgroups of S_6 are isomorphic to?
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  3. #3
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    yes

    !
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rtrt1 View Post
    !
    Okay, well what is the order of a Sylow-3 subgroup of S_6? How many groups are there of this order? (HINT: there are only two. Why?)

    You now just have to work out which of these two groups a Sylow-3 group is isomorphic to. This reduces to proving whether or not there exists an element of order 9 in your group (why?).

    So, does there exist an element of order 9 in S_6?

    EDIT: Alternatively, you can go back to the previous time you posted this question, here, just over two weeks ago, and do what I suggested then. That is, look at the group generated by (123) and (456). What is its order (from it's order you can tell if it is a Sylow-3 subgroup)? What group is it isomorphic to? (It's not cyclic, but it is abelian).
    Last edited by Swlabr; July 28th 2010 at 05:55 AM.
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  5. #5
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    so

    I understand that this sub group is not isomorphous to Z3XZ3
    So which group is isomorphous?
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rtrt1 View Post
    I understand that this sub group is not isomorphous to Z3XZ3
    So which group is isomorphous?
    Why is it not isomorphic to \mathbb{Z}_3 \times \mathbb{Z}_3?
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  7. #7
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    Because

    Because it is not cyclic ( but Z3 is Cyclic )
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  8. #8
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rtrt1 View Post
    Because it is not cyclic ( but Z3 is Cyclic )
    Yes, \mathbb{Z}_3 is cyclic, but \mathbb{Z}_3 \times \mathbb{Z}_3 is not ( V_4=K_4, the Klein 4-group, is not cyclic, and is isomorphic to \mathbb{Z}_2 \times \mathbb{Z}_3).
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  9. #9
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    so

    So that the group I was looking for ...
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  10. #10
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rtrt1 View Post
    So that the group I was looking for ...
    Yup.
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