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

TL;DR

This paper investigates the structure of the Gruenberg-Kegel graph of finite solvable groups, establishing bounds on the Fitting length when the graph has cut-vertices or cut-sets, and describing the graph's structure in specific cases.

Contribution

It proves that solvable groups with a cut-set in their Gruenberg-Kegel graph have a controlled series length, bounds the Fitting length with minimal cut-sets, and characterizes the graph when minimal cut-sets of size two exist.

Findings

Bound on the Fitting length for groups with a cut-vertex in the graph.

Existence of a $ ext{sigma}$-series of length 5 for groups with a cut-set.

Geometrical description of the graph when it has a minimal cut-set of size 2.

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 $G$.