Group Orders That Imply a Nontrivial p-Core

Given a prime number $p$ and a natural number $m$ not divided by $p$, we propose the problem of finding the smallest number $r_{0}$ such that for $r\geq r_{0}$, every group $G$ of order $p^{r}m$ has a non-trivial normal $p$-subgroup. We prove that we can explicitly calculate the number $r_{0}$ in the case where every group of order $p^{r}m$ is solvable for all $r$, and we obtain the value of $r_{0}$ for a case where $m$ is a product of two primes.