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!
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.
