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Math Help - p-Sylow question

  1. #1
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    p-Sylow question

    Let K be a normal subgroup of a fintire group G. Let S be a p-Sylow subgroup of G (p a prime divisor of |G|).
    Prove that KS/K is a p-Sylow subgroup of G/K.
    Proof/
    So we know that K is a subgroup of KS and KS is a subgroup of G, so by the correspondence theroem KS/K is a subgroup of G/K. Since |KS/K|=|S|/|K intersect S|, then |KS/K|= p (where p is a prime).
    Want to show that KS/K is a miaximal p-subgroup of G/K.
    So it suffices to show that [G/K:KS/K] is relatively prime to p. From here is where I'am having trouble I have been trying to use the facts about S being a p-sylow subgroup of G, to get that S/K is a p-sylow subgroup of G/K to try to get my conclusion,but I have been getting no where doing this. Any suggestions would be a life saver.
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  2. #2
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    Quote Originally Posted by aabsdr View Post
    Let K be a normal subgroup of a fintire group G. Let S be a p-Sylow subgroup of G (p a prime divisor of |G|).
    Prove that KS/K is a p-Sylow subgroup of G/K.
    Proof/
    So we know that K is a subgroup of KS and KS is a subgroup of G, so by the correspondence theroem KS/K is a subgroup of G/K. Since |KS/K|=|S|/|K intersect S|, then |KS/K|= p (where p is a prime).
    Want to show that KS/K is a miaximal p-subgroup of G/K.
    So it suffices to show that [G/K:KS/K] is relatively prime to p. From here is where I'am having trouble I have been trying to use the facts about S being a p-sylow subgroup of G, to get that S/K is a p-sylow subgroup of G/K to try to get my conclusion,but I have been getting no where doing this. Any suggestions would be a life saver.

    [G/K:KS/K]=\frac{\frac{|G|}{|K|}}{\frac{|KS}{|K|}} =\frac{|G|}{|KS|}=\frac{|G||K\cap S|}{|K||S|} . If we put |G|=p^rm\,,\,\,(p,m)=1 , we get:

    [G/K:KS/K]=\frac{|G||K\cap S|}{|K||S|}=\frac{p^rm\cdot |K\cap S|}{|K|p^r}=\frac{m|K\cap S|}{|K|} , and now we just have to note that any power of p in |K| cancels with the same power of p in |K\cap S| (why??)

    Tonio
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  3. #3
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    any power of p in |K| cancels with the same power of P in |K intersect S| because K intersect S is a maximal P subgroup of K.
    Thanks you sooooooo much!!!!!!!!!!!!!!!!!!!!
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