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Sylow numbers and nilpotent Hall subgroups. (English) Zbl 1285.20019
Let $$\pi$$ be a set of primes of a finite group $$G$$. By the renowned Wielandt’s theorem, if $$G$$ has a nilpotent Hall $$\pi$$-subgroup then all Hall $$\pi$$-subgroups of $$G$$ are conjugate and every $$\pi$$-subgroup is contained in some Hall $$\pi$$-subgroup of $$G$$. In particular, all $$\pi$$-subgroups of $$G$$ are nilpotent. The author presents a very interesting criterion for a finite group to possess nilpotent Hall $$\pi$$-subgroups. This is done by means of the Sylow numbers.
The main result claims that $$G$$ has nilpotent Hall $$\pi$$-subgroups if and only if the three following conditions hold: (i) For any two different primes $$p,q\in\pi$$, the prime $$p$$ does not divide the number of Sylow $$q$$-subgroups of $$G$$. (ii) If $$\{2,3\}\subseteq\pi$$, then $$G$$ does not have any composition factor isomorphic to $$\text{PSL}(2,q)$$ with $$(q^2-1)_{\{2,3\}}=24$$. (iii) If $$\{2,7\}\subseteq\pi$$, then $$G$$ does not have any composition factor isomorphic to $$^2G_2(3^{2n+1})$$ with $$n\not\equiv 3\pmod 7$$. – The proof is made first for the case $$|\pi|=2$$ and then for the general case, and it depends on the recently completed classification of Hall subgroups of simple groups.

##### MSC:
 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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