Hi: Problem: Let U be a subgroup of G and 1 < |G:U| <5. Then |G| < 4 or G is not simple.
If |G:U| = 2 I can use a problem (problem A) that says: Let p be the smallest prime divisor of |G|. Then every subgroup of index p is normal in G. That is because in this case, 2 is a divisor of |G| and it is the smallest prime. For case |G:U| = 3 if |G| is not even then, by problem A I am done. If |G| is even, however, problem A is of no use. The same for |G:U| = 4. Any hint?