Products of \(\mathrm{F}^\ast(G)\)-subnormal subgroups of finite groups. (English. Russian original) Zbl 1381.20021
Russ. Math. 61, No. 6, 66-71 (2017); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2017, No. 6, 76-82 (2017).
Summary: A subgroup \(H\) of a finite group \(G\) is called \(\mathrm{F}^\ast(G)\)-subnormal if \(H\) is subnormal in \(H\mathrm{F}^\ast(G)\). We show that if a group \(G\) is a product of two \(\mathrm{F}^\ast(G)\)-subnormal quasinilpotent subgroups, then \(G\) is quasinilpotent. We also study groups \(G = AB\), where \(A\) is a nilpotent \(\mathrm{F}^\ast(G)\)-subnormal subgroup and \(B\) is a \(\mathrm{F}^\ast(G)\)-subnormal supersoluble subgroup. Particularly, we show that such groups \(G\) are soluble.
MSC:
| 20D35 | Subnormal subgroups of abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D40 | Products of subgroups of abstract finite groups |
Keywords:
finite group; \(\mathrm{F}^\ast(G)\)-subnormal subgroup; nilpotent group; supersoluble group; quasinilpotent group; product of subgroupsReferences:
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