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Normal subgroup existence

  1. Mar 21, 2010 #1
    G,H be groups(finite or infinite)
    Prove that if (G:H)=n, then there exist some normal subgroup K of G (G:K)≤n!
    example) let G=A5, H=A4 then (G:H)=5, then K={id} exists, (G:K)≤5!
     
  2. jcsd
  3. Mar 21, 2010 #2
    This is a standard result. If you think how elements of G can act on the left (or right) cosets of H you should come up with a homomorphism of G into the group of permutations of the cosets. Then think about the kernel of the homomorphism.
     
  4. Mar 21, 2010 #3
    Actually you have (G:K)|n! which is sometimes more useful.
     
  5. Mar 22, 2010 #4
    Thanks a lot Martin. I understand it.
     
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