**Notation and Remark. **Let be a finite group** **and let be a prime number which divides We let be the number of Sylow -subgroups of Note that if a Sylow -subgroup is not normal, then because and So if is simple, then for all primes

**Problem**. Prove that a group with and where are primes, is not simple.

**Solution**. If then and we know that every group of order where is a prime and has a subgroup of order and all such subgroups are normal. So in this case is not simple. Thus we may assume that The case is obvious. So we will consider two cases:

*Case 1*. : [we do not need the condition for this case.] Suppose that is simple. Then and either or If then will have elements of order because every Sylow -subgroup of has exatly elements of order But then we will only have elements left and so which is a contradiction. If then and which is nonsense.

*Case 2*. : Suppose that is simple. Let be two -Sylow subgroups of and put Then

Thus Therefore, since and we’ll get So is a normal subgroup of and Hence contains both and so As a result for some But then

which gives us Since the only possibility would be Thus i.e.