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Math Help - Sylow p-subgroup

  1. #1
    Junior Member ibnashraf's Avatar
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    Sylow p-subgroup

    can someone explain what is meant by a Sylow p-subgroup

    and hence outline the steps in the following question:

    "Let G be a group. Prove that if G has only one Sylow p-subgroup H, then H is normal in G."
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    Junior Member ibnashraf's Avatar
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    Re: Sylow p-subgroup

    Quote Originally Posted by ibnashraf View Post
    "Let G be a group. Prove that if G has only one Sylow p-subgroup H, then H is normal in G."
    I think one of my problems here is that G was defined to be a group.
    If G was defined to be a "finite group", then I could have said that by definition H is a subgroup of G.
    However, there is no finite group mentioned in the question, so how exactly am I to proceed ?
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    Re: Sylow p-subgroup

    Quote Originally Posted by ibnashraf View Post
    I think one of my problems here is that G was defined to be a group.
    If G was defined to be a "finite group", then I could have said that by definition H is a subgroup of G.
    What? H is given as a subgroup so that is not in question.

    However, there is no finite group mentioned in the question, so how exactly am I to proceed ?
    The problem is to show that H is a normal subgroup. So what is the definition of "normal subgroup"?
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    Junior Member ibnashraf's Avatar
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    Re: Sylow p-subgroup

    Quote Originally Posted by HallsofIvy View Post
    What? H is given as a subgroup so that is not in question.


    The problem is to show that H is a normal subgroup. So what is the definition of "normal subgroup"?
    ok well i guess since H is already a subgroup then i have to show that g^{-1}hg \in H\Rightarrow H is normal in G ????
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    Re: Sylow p-subgroup

    yes, you have to show that for any g in G, and h in H, that ghg-1 is in H.

    equivalently (and easier to use for this problem), you need to show that gHg-1 is a subset of H, for any g in G.

    some things to think about:

    a) prove that for any subgroup H of G, and any element g in G, the set gHg-1 = {ghg-1: h in H} is actually a subgroup of G.

    b) show that H and gHg-1 have the same order.

    conclude that if G has only ONE subgroup of a given order, that subgroup MUST be normal (sylow subgroup or not).

    the definition of a sylow p-subgroup is, by the way:

    a subgroup of order pk, where pk divides |G|, and pk+1 does not divide |G|.

    for example, if G is a group of order 24, a sylow 2-subgroup of G is a subgroup of order 8, whereas a sylow 3-subgroup is a subgroup of order 3.
    Last edited by Deveno; June 18th 2012 at 05:58 AM.
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