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Abstract-Sylow Theorems

  1. Dec 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Let G be a finite group of order n and let p be a prime number that divides n.
    Let's mark as Op(G) as the intersection of all p-sylow groups of G.

    1. Prove that Op(G) is a normal p-subgroup in G. -I've managed to prove it..
    2. Prove that every normal p-subgroup of G is contained in Op(G) -I've managed to prove this also...
    3. Let's mark: G] = G/Op(G). Prove that Op(G] ) ={1} -I have no idea about this...
    Help is needed :(

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Dec 15, 2009 #2
    Use the correspondence between subgroups of [tex]G/O_p(G)[/tex] and subgroups of [tex]G[/tex] containing [tex]O_p(G)[/tex].
  4. Dec 15, 2009 #3
    I've managed to prove it on my own...TNX anyway
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