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

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 \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 $\Gamma (G)$ when it has a minimal cut-set of size $2$, for a finite solvable group $G$.

arXiv commentThis a substatially revised form of "Finite solvable groups whose Gruenberg-Kegel graph has a cut set" to appear in the Czechoslovak Mathematical Journal