Groups whose proper subgroups are Baer groups. (English) Zbl 0840.20030
The author considers minimal non-nilpotent and minimal non-Baer groups (MNN-groups, MNB-groups). The main results: (1) An infinite soluble MNN-groups possesses maximal subgroups or all subgroups are subnormal (Theorem 2.5), (2) An infinite soluble group \(G\) is an MNB-group if and only if it possesses a normal subgroup \(N\) such that (a) \(G/N\) is an MNN-group with maximal subgroups, (b) \(\langle x,N\rangle\) is a Baer-group for every \(x\) of \(G\), and (c) \(HN=G\) implies \(H=G\). An example of an MNB-group is given that is not an MNN-group, it contains a subgroup of index 2 which is non-hypercentral and satisfies the normalizer condition.
Reviewer: H.Heineken (Würzburg)
MSC:
| 20F19 | Generalizations of solvable and nilpotent groups |
| 20E15 | Chains and lattices of subgroups, subnormal subgroups |
| 20E28 | Maximal subgroups |
| 20E07 | Subgroup theorems; subgroup growth |
Keywords:
minimal non-nilpotent non-Baer groups; MNN-groups; minimal non-Baer groups; MNB-groups; infinite soluble MNN-groups; maximal subgroupsReferences:
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