Normal subgroup?

Question

Show if a finite group G has a subgroup H of index n,then H contains a normal subgroup of G of index a divisor of n factorial

Answer 1

Do your own homework

Answer 2

There is a homomorphism from G into the full permutation group on the n cosets of G/H. The order of the image of G under the homomorphism is a divisor of the order of the full permutation group on n elements, which is n! But that number is also the index in G of the kernel of the homomorphism, which is a normal subgroup of G.