A criterion of \(p\)-hypercyclically embedded subgroups of finite groups. (English) Zbl 1300.20030
Summary: Let \(G\) be a finite group, \(E\) a normal subgroup of \(G\) and \(p\) a fixed prime. We say that \(E\) is \(p\)-hypercyclically embedded in \(G\) if every \(p\)-\(G\)-chief factor of \(E\) is cyclic. A subgroup \(H\) of \(G\) is said to satisfy \(\Pi\)-property in \(G\) if \(|G/K:N_{G/K}((H\cap L)K/K)|\) is a \(\pi((H\cap L)K/K)\)-number for any chief factor \(L/K\) in \(G\); \(H\) is said to be \(\Pi\)-normal in \(G\) if \(G\) has a subnormal supplement \(T\) of \(H\) such that \(H\cap T\leqslant I\leqslant H\) for some subgroup \(I\) satisfying \(\Pi\)-property in \(G\). In this paper, we prove that \(E\) is \(p\)-hypercyclically embedded in \(G\) if and only if some classes of \(p\)-subgroups of \(E\) are \(\Pi\)-normal in \(G\). Many recent results are extended.
MSC:
| 20D40 | Products of subgroups of abstract finite groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
| 20D30 | Series and lattices of subgroups |
Keywords:
finite groups; \(p\)-hypercyclically embedded subgroups; subnormal supplements; chief factors; \(\Pi\)-property; \(\Pi\)-normality; \(p\)-subgroupsReferences:
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