Czechoslovak Mathematical Journal, Vol. 75, No. 4, pp. 1093-1104, 2025
Finite solvable groups whose Gruenberg-Kegel graph has a cut-set
Lorenzo Bonazzi
Received October 7, 2021. Published online November 21, 2025.
Abstract: 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 $2$.
Keywords: Gruenberg-Kegel graph; prime graph of finite group; cut-set; solvable group
Affiliations: Lorenzo Bonazzi, Department of Mathematics and Informatics, University of Florence, Viale Morgagni 67/a, 50134 Florence, Italy, e-mail: lorenzo.bonazzi@unifi.it
