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On a theorem of Frobenius. (English) Zbl 0994.20016

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 \(b\).

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D40 Products of subgroups of abstract finite groups
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