Finite groups with abnormal and \(\mathfrak{U}\)-subnormal subgroups. (English. Russian original) Zbl 1384.20016
Sib. Math. J. 57, No. 2, 352-363 (2016); translation from Sib. Mat. Zh. 57, No. 2, 447-462 (2016).
All groups considered in the paper under review are finite. Here, the author studies mainly two problems. The first problem is to study the structure of a group every nilpotent subgroup of which is \({\mathcal U}\)-subnormal or \({\mathcal U}\)-abnormal, where \({\mathcal U}\) denotes the formation of all supersoluble groups. The second problem is to describe the groups every nilpotent subgroup of which is abnormal or \(\mathbb{P}\)-subnormal, where \(\mathbb{P}\) denotes the set of all prime integers. The author obtaines the necessary and sufficient conditions for these groups. In particular, it is proved that they have a Sylow towers. Also, the author checks that Problem 20 in L. A. Shemetkov’s book [Формации конечных групп (Russian). Moscow: Nauka Publishers. (1978; Zbl 0496.20014)] has a negative solution.
Reviewer: Bui Xuan Hai (Ho Chi Minh City)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D35 | Subnormal subgroups of abstract finite groups |
Citations:
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