Is There a Normal Subgroup K in Groups G and H with Index (G:K) ≤ n!?

[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!

 

[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.

 

[Martin Rattigan]

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

 

[emptyboat]

Thanks a lot Martin. I understand it.