# Normal subgroup existence

1. Mar 21, 2010

### emptyboat

G,H be groups(finite or infinite)
Prove that if (G)=n, then there exist some normal subgroup K of G (G:K)≤n!
example) let G=A5, H=A4 then (G)=5, then K={id} exists, (G:K)≤5!

2. Mar 21, 2010

### Martin Rattigan

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. Mar 21, 2010

### Martin Rattigan

Actually you have (G:K)|n! which is sometimes more useful.

4. Mar 22, 2010

### emptyboat

Thanks a lot Martin. I understand it.