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On cover-avoiding subgroups of Sylow subgroups of finite groups. (English) Zbl 1197.20014

From the introduction: A subgroup \(H\) of a finite group \(G\) is said to have the cover-avoidance property in \(G\), \(H\) is a CAP subgroup of \(G\) in short, if \(H\) either covers or avoids every chief factor of \(G\). In the present work, we fix a subgroup \(D\) in every Sylow subgroup \(P\) of \(F^*(G)\) satisfying \(1<|D|<|P|\) and study the structure of \(G\) under the assumption that every subgroup \(H\) with \(|H|=|D|\) has the cover-avoidance property in \(G\). We state our results in the broader context of formation theory.
Main Theorem. Let \(\mathcal F\) be a saturated formation containing \(\mathcal U\), the class of all supersolvable groups, and \(E\) a normal subgroup of \(G\) such that \(G/E\in\mathcal F\). Suppose that, for every non-cyclic Sylow subgroup \(P\) of \(F^*(E)\), \(P\) has a subgroup \(D\) such that \(1<|D|<|P|\) and all subgroups \(H\) of \(P\) with order \(|H|=|D|\) and with order \(|H|=2|D|\) (if \(P\) is a non-Abelian 2-group and \(|P:D|>2\)) satisfy the cover-avoidance property in \(G\). Then \(G\in\mathcal F\).

MSC:

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