On the cut-set of the Gruenberg-Kegel graph of a finite solvable group
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913808549675008 |
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| author | Bonazzi, Lorenzo |
| author_facet | Bonazzi, Lorenzo |
| contents | Let $Γ(G)$ be the Gruenberg-Kegel graph of a finite group $G$. We prove that if $G$ is solvable and $σ$ is a cut-set for $Γ(G)$, then $G$ has a $σ$-series of length $5$ whose factors are controlled. As a consequence, we prove that if $G$ is a solvable group and $Γ(G)$ 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 \emph{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 $Γ(G)$ when it has a minimal cut-set of size $2$, for a finite solvable group $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_03723 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the cut-set of the Gruenberg-Kegel graph of a finite solvable group Bonazzi, Lorenzo Group Theory 20D60 Let $Γ(G)$ be the Gruenberg-Kegel graph of a finite group $G$. We prove that if $G$ is solvable and $σ$ is a cut-set for $Γ(G)$, then $G$ has a $σ$-series of length $5$ whose factors are controlled. As a consequence, we prove that if $G$ is a solvable group and $Γ(G)$ 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 \emph{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 $Γ(G)$ when it has a minimal cut-set of size $2$, for a finite solvable group $G$. |
| title | On the cut-set of the Gruenberg-Kegel graph of a finite solvable group |
| topic | Group Theory 20D60 |
| url | https://arxiv.org/abs/2110.03723 |