Given a prime number p and a natural number m not divisible by p, we propose the problem of finding the smallest number r0 such that for r≥r0, every group G of order prm has a non-trivial normal p-subgroup. We prove that we can explicitly calculate the number r0 in the case where every group of order prm is solvable for all r, and we obtain the value of r0 for a case where m is a product of two primes.

Keywords: p-core, normal subgroups