Article
| Title: | Finite solvable groups whose Gruenberg-Kegel graph has a cut-set (English) |
| Author: | Bonazzi, Lorenzo |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1093-1104 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $\Gamma (G)$ be the Gruenberg-Kegel graph of a finite group $G$. We prove that if $G$ is solvable and $\sigma $ is a cut-set for $\Gamma (G)$, then $G$ has a $\sigma $-series of length 5 whose factors are controlled. As a consequence, we prove that if $G$ is a solvable group and $\Gamma (G)$ has a cut-vertex $p$, then the Fitting length $\ell _F(G)$ of $G$ is bounded and the bound obtained is the best possible. A cut-set is said minimal if it does not contain any other proper subset that is a cut-set for the graph. For a finite solvable group $G$, we give a geometrical description of $\Gamma (G)$ when it has minimal cut-set of size 2. (English) |
| Keyword: | Gruenberg-Kegel graph |
| Keyword: | prime graph of finite group |
| Keyword: | cut-set |
| Keyword: | solvable group |
| MSC: | 20F16 |
| DOI: | 10.21136/CMJ.2025.0380-21 |
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| Date available: | 2025-12-20T07:04:49Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153231 |
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| Reference: | [4] Gruber, A., Keller, T. M., Lewis, M. L., Naughton, K., Strasser, B.: A characterization of the prime graph of solvable groups.J. Algebra 442 (2015), 397-422. Zbl 1331.20029, MR 3395066, 10.1016/j.jalgebra.2014.08.040 |
| Reference: | [5] Isaacs, I. M.: Character Theory of Finite Groups.Pure and Applied Mathematics 69. Academic Press, New York (1976). Zbl 0337.20005, MR 0460423, 10.1016/s0079-8169(08)x6037-4 |
| Reference: | [6] Lucido, M. S.: The diameter of a prime graph of a finite group.J. Group Theory 2 (1999), 157-172 \99999DOI99999 10.1515/jgth.1999.011 . Zbl 0921.20020, MR 1681526, 10.1515/jgth.1999.011 |
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