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On \(c\)-normality of certain subgroups of prime power order of finite groups. (English) Zbl 1082.20008

Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called \(c\)-normal in \(G\) if \(G\) has a normal subgroup \(N\) such that \(G=NH\) and \(N\cap H\leq\text{Core}(H)\), where \(\text{Core}(H)\) is the intersection of all the conjugates of \(H\) in \(G\).
The authors prove the following main results: Let \(\mathcal F\) be a saturated formation containing the formation of supersoluble groups. Then \(G\in\mathcal F\) if and only if there is a normal subgroup \(H\) of \(G\) such that \(G/H\in\mathcal F\) and all maximal subgroups of the Sylow subgroups of \(H\) are \(c\)-normal in \(G\) (Theorem 3.3). The same result holds if we replace the maximal subgroups of the Sylow subgroups of \(H\) by the subgroups of prime order or order \(4\) of \(H\) (Theorem 3.9).
The proofs of the above results rely on the following \(p\)-supersolubility criterion: Let \(G\) be a finite \(p\)-soluble group for a prime \(p\). Then \(G\) is \(p\)-supersoluble if there is a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-supersoluble and either all maximal subgroups of the Sylow subgroups of \(H\) are \(c\)-normal in \(G\) or the subgroups of prime order or order \(4\) of \(H\) are \(c\)-normal in \(G\) (Theorems 3.1 and 3.7).

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
20E28 Maximal subgroups
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