Finite groups with \(X\)-quasipermutable subgroups of prime power order. (English) Zbl 1373.20036
Summary: Let \(H\), \(L\) and \(X\) be subgroups of a finite group \(G\). Then \(H\) is said to be \(X\)-permutable with \(L\) if for some \(x\in X\) we have \(AL^{x}=L^{x}A\). We say that \(H\) is \(X\)-quasipermutable (\(X_{S}\)-quasipermutable, respectively) in \(G\) provided \(G\) has a subgroup \(B\) such that \(G=N_{G}(H)B\) and \(H\) \(X\)-permutes with \(B\) and with all subgroups (with all Sylow subgroups, respectively) \(V\) of \(B\) such that \((|H|, |V|)=1\). In this paper, we analyze the influence of \(X\)-quasipermutable and \(X_{S}\)-quasipermutable subgroups on the structure of \(G\). Some known results are generalized.
MSC:
| 20D40 | Products of subgroups of abstract finite groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20E07 | Subgroup theorems; subgroup growth |
