On a theorem of Frobenius (Q2774613)

Let \(n\) and \(m\) be positive integers and let \(p\) be a prime. The integer \(n\) has the \((p,m)\)-property if for each prime divisor \(q\) of \(n\), \(q\) does not divide \(p^k-1\), for \(k=1,\dots,m\). Applying the Frobenius criterion for \(p\)-nilpotency, the author proves: If \(G\) is a finite solvable group such that \(|G|=ab\), \((a,b)=1\), and \(a\) has the \((p,d)\)-property for any prime \(p\) dividing \(b\) and \(d\) is the maximal exponent of \(p\) in \(b\), then \(G\) is a semidirect product of two normal Hall subgroups, one of order \(a\) and one of order \(b\).