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Math Help - Sylow subgroup question

  1. #1
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    Sylow subgroup question

    If H is a normal subgroup of a finite group G and |H|=p^K for some prime p, show H is contained in every p-Sylow subgroup of G. I am totally lost on this one, if anyone could help me it would be great
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  2. #2
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    Quote Originally Posted by aabsdr View Post
    If H is a normal subgroup of a finite group G and |H|=p^K for some prime p, show H is contained in every p-Sylow subgroup of G. I am totally lost on this one, if anyone could help me it would be great
    Hint: Any two Sylow-p subgroups of G are conjugate, and H is a subgroup of G, which is invariant under conjugation by members of G (see here).
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  3. #3
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    so here is what I have so far:
    Let G be a finite group with H a normal subgroup of G such that |G|=p^k for some prime p. By Sylow's second theroem, H is contained in some Sylow p-subgroup of G. Sylow's third thereom says that any two Sylow p-subgroups of G are conjugate.
    I am not sure after this, is that any two Sylow 0-subgroups being conjugate enough to show that H is contanied in ever p_Sylow subgroup of G since we know it is contained in one of the p-Sylow subgroups of G?
    Sorry i am kind of slow on some of this stuff, i really appreciate the help.
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  4. #4
    Senior Member roninpro's Avatar
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    Suppose H\subseteq P, where P is some p-Sylow subgroup. Then, if Q is another p-Sylow subgroup, we can find g\in G so that Q=gPg^{-1}. Then, we know that gHg^{-1}\subseteq Q, but what is this?
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  5. #5
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    Then, we know that gHg^{-1}\subseteq Q, but what is this?

    So H conjugate is contained in Q, but I am not sure what that is telling me. Could you explain a little more please?
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  6. #6
    Senior Member roninpro's Avatar
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    You know that H is normal in G. What can you say about gHg^{-1}?
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  7. #7
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    You know that H is normal in G. What can you say about gHg^{-1}?
    gHg^{-1} is normal in G, right?
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  8. #8
    Senior Member roninpro's Avatar
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    Yes, but you can say something a lot more specific.

    What does it even mean to be normal in G?
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  9. #9
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    If H is a normal subgroup of G than aH=Ha for all a in G. so aH(a^-1)=H, which means that if H conjugate is contained in Q, than H is contained in Q, right? So than we can apply this to every p-Sylow subgroup of G to show that a normal subgroup H of G is contained in every p-Sylow subgroup of G?
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  10. #10
    Senior Member roninpro's Avatar
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    Exactly. You got it.
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  11. #11
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    thanks so much for your help, I was really struggle with this as you could tell
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