Question

Let |G| = n < ∞, H be a subgroup of G, and assume that the number n does not divide the factorial of the index of H in G.

Prove that there exists a normal subgroup N ≠ {e} of G contained in H.

Answer 1

(Hint: Consider the natural left action of the group G on the left coset space X = G/H.)

Answer 2

sad to say i just saw a proof of this on line, but of course i forget it
google it, pretty sure it will come up

Answer 3

thanks