On \(\sigma\)-supersoluble groups and one generalization of CLT-groups. (English) Zbl 1439.20013
In [J. Algebra 436, 1–16 (2015; Zbl 1316.20020)], A. N. Skiba introduced the concept of \(\sigma\)-nilpotent groups and \(\sigma\)-soluble groups. In this article, the authors introduced the concept of \(\sigma\)-supersoluble groups. Let \(G\) be a finite group and \(G^{\mathfrak{N}_{\sigma}}\) be the \(\sigma\)-nilpotent residual of \(G\), that is, the intersection of all normal subgroups \(N\) of \(G\) with \(\sigma\)-nilpotent quotient \(G/N\). Then we say that \(G\) is \(\sigma\)-supersoluble if each chief factor of \(G\) below \(G^{\mathfrak{N}_{\sigma}}\) is cyclic. The authors investigated some interesting properties of \(\sigma\)-supersoluble groups and obtained some applications of such groups in the theory of generalized \(CLT\)-groups.
Reviewer: Haoran Yu (Changchun)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D30 | Series and lattices of subgroups |
Keywords:
finite group; \(\sigma\)-nilpotent group; \(\sigma\)-soluble group; \(\sigma\)-supersoluble group; \(CLT_\sigma\)-groupCitations:
Zbl 1316.20020References:
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