A generalization of cover-avoiding properties in finite groups. (English) Zbl 1227.20016
Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called a CAP- (CAP*-)subgroup of \(G\) if each chief factor (non-Frattini chief factor) of \(G\) is either covered or avoided by \(H\). The family of all maximal subgroups of all Sylow subgroups of \(G\) is denoted by \(\mathcal M(G)\).
If \(H\) is a normal subgroup of \(G\) such that every member of \(\mathcal M(H)\) is a CAP-subgroup of \(G\), then every chief factor of \(G\) contained in \(H\) is cyclic (Theorem 1.6). If \(\mathcal F\) is a saturated formation containing the class of supersoluble groups and \(H\) is a normal subgroup of \(G\) such that \(G/H\in\mathcal F\) and every member of \(\mathcal M(H)\) is a CAP*-subgroup of \(G\), then \(G\in\mathcal F\) (Theorem 1.7). Moreover, several well-known theorems on CAP-subgroups are improved in the present paper (Section 4).
If \(H\) is a normal subgroup of \(G\) such that every member of \(\mathcal M(H)\) is a CAP-subgroup of \(G\), then every chief factor of \(G\) contained in \(H\) is cyclic (Theorem 1.6). If \(\mathcal F\) is a saturated formation containing the class of supersoluble groups and \(H\) is a normal subgroup of \(G\) such that \(G/H\in\mathcal F\) and every member of \(\mathcal M(H)\) is a CAP*-subgroup of \(G\), then \(G\in\mathcal F\) (Theorem 1.7). Moreover, several well-known theorems on CAP-subgroups are improved in the present paper (Section 4).
Reviewer: Hans Lausch (Clayton)
MSC:
| 20D30 | Series and lattices of subgroups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
Keywords:
CAP-subgroups, CAP*-subgroups; saturated formations; Frattini chief factors; cover-avoiding property; finite supersoluble groupsReferences:
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