Normal subgroup existence

  • Thread starter emptyboat
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  • #1
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Main Question or Discussion Point

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!
 

Answers and Replies

  • #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.
 
  • #3
Actually you have (G:K)|n! which is sometimes more useful.
 
  • #4
21
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Thanks a lot Martin. I understand it.
 

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