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Frattini argument for Hall subgroups. (English) Zbl 1316.20019

Let \(G\) be a finite group. The often applied method Frattini argument has many extensions such as that every normal subgroup \(A\) of \(G\) possesses a maximal solvable subgroup \(S\) such that \(G=AN_G(S)\) by V. I. Zenkov et al. [Algebra Logika 43, No. 2, 184-196 (2004); translation in Algebra Logic 43, No. 2, 102-108 (2004; Zbl 1079.20035)].
Let \(\pi\) be a set of primes, \(\text{Hall}_\pi(G)\) the set of all \(\pi\)-Hall subgroups of \(G\). Let \(E_\pi\) be the class of all \(G\) with \(\text{Hall}_\pi(G)\neq\emptyset\), \(C_\pi\) be the class of all \(E_\pi\)-groups \(G\) such that \(\pi\)-Hall subgroups are conjugate. The note that if \(A\) is a normal \(C_\pi\)-subgroup of \(G\) then \(G=AN_G(H)\) for every \(H\in\text{Hall}_\pi(A)\) is important in the various extensions of Sylow’s theorems surveyed in the authors’ [Russ. Math. Surv. 66, No. 5, 829-870 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 3-46 (2011; Zbl 1243.20027)].
The main result of the paper is the following. Let \(A\) be a normal subgroup of the \(E_\pi\)-group \(G\). Then there exists \(H\in\text{Hall}_\pi(A)\) with \(G=AN_G(H)\), and \(N_G(H)\) is an \(E_\pi\)-group such that \(\text{Hall}_\pi(N_G(H))\subset\text{Hall}_\pi(G)\).

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D25 Special subgroups (Frattini, Fitting, etc.)
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D30 Series and lattices of subgroups
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References:

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