Group Orders That Imply a Nontrivial p-Core

Abstract

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.