Finite solvable groups whose Gruenberg-Kegel graph has a cut-set
Year: 2025
Type: article
Abstract: 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.
Author: Lorenzo Bonazzi
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3205351459