## 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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