A note on TI-subgroups of a finite group. (English) Zbl 1291.20018
Summary: Let \(G\) be a finite group and \(H\) a subgroup of \(G\). Recall that \(H\) is said to be a TI-subgroup of \(G\) if \(H^g\cap H=1\) or \(H\) for each \(g\in G\). In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group \(G\) are TI-subgroups, then \(G\) is soluble, and all non-nilpotent subgroups of \(G\) are normal.
MSC:
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
References:
| [1] | DOI: 10.1016/j.jalgebra.2006.10.001 · Zbl 1116.20014 · doi:10.1016/j.jalgebra.2006.10.001 |
| [2] | Li S., Math. Proc. R. Ir. Acad. 100 pp 65– (2000) |
| [3] | Shi J., J. Algebra Appl. 12 (2013) |
| [4] | DOI: 10.1007/BF01238459 · Zbl 0388.20011 · doi:10.1007/BF01238459 |
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