Groups with almost Frattini closed subgroups. (English) Zbl 1526.20043
Summary: A subgroup \(H\) of a group \(G\) is said to be Frattini closed in \(G\) if either \(H=G\) or \(H\) is the intersection of all maximal subgroups of \(G\) containing \(H\). The structure of a soluble group in which every subgroup is Frattini closed is known. In this paper, the behavior of a (generalized) soluble group \(G\) in which every subgroup is Frattini closed in a subgroup of finite index of \(G\) is studied. Among other results, it is proved that if \(G\) is a (generalized) soluble group and there exists a positive integer \(k\) such that every subgroup of \(G\) is Frattini closed in a subgroup of index at most \(k\) in \(G\), then \(G\) contains a normal subgroup of finite index in which all subgroups are Frattini closed.
MSC:
| 20E28 | Maximal subgroups |
| 20F19 | Generalizations of solvable and nilpotent groups |
| 20E15 | Chains and lattices of subgroups, subnormal subgroups |
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