On the cut-set of the Gruenberg-Kegel graph of a finite solvable group

Abstract

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 $G$.