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Finite groups with \(X\)-quasipermutable Sylow subgroups. (English) Zbl 1384.20020

Ukr. Math. J. 67, No. 12, 1941-1950 (2016) and Ukr. Mat. Zh. 67, No. 12, 1715-1722 (2015).
Summary: Let \(H\leq E\) and \(X\) be subgroups of a finite group \(G\). Then we say that \(H\) is \(X\)-quasipermutable (\(X_S\)-quasipermutable, respectively) in \(E\) provided that \(G\) has a subgroup \(B\) such that \(E = N_E(H)B\) and \(H\) is \(X\)-permutable with \(B\) and with all subgroups (with all Sylow subgroups, respectively) \(V\) of \(B\) such that \((|H|, |V|) = 1\). We analyze the influence of \(X\)-quasipermutable and \(X_S\)-quasipermutable subgroups on the structure of \(G\). In particular, it is proved that if every Sylow subgroup \(P\) of \(G\) is \(F(G)\)-quasipermutable in its normal closure \(P^G\) in \(G\), then \(G\) is supersoluble.

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D15 Finite nilpotent groups, \(p\)-groups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D40 Products of subgroups of abstract finite groups
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References:

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