On the cut-set of the Gruenberg-Kegel graph of a finite solvable group
Lorenzo Bonazzi
TL;DR
The paper refines the Gruenberg-Kegel framework for finite solvable groups by proving that whenever $\sigma$ is a cut-set for the Gruenberg-Kegel graph $\Gamma(G)$ of a solvable group $G$, there exists a normal $\sigma$-series of length $5$ with a prescribed structure on the factors. As a corollary, if $\Gamma(G)$ has a cut-vertex $p$, the Fitting length $\ell_F(G)$ is bounded (with the bound shown to be best possible), and a minimal cut-set of size $2$ yields a geometric description of $\Gamma(G)$. The authors develop a detailed decomposition using $\pi_1(G)$ and $\pi_2(G)$ associated to Hall subgroups, and in the exceptional case $G/{\bf O}(G)\simeq (2.S_4)^-$, the non-nilpotent quotient is explicitly controlled ($h(G/G_3)=2$). The paper also constructs explicit examples to show the sharpness of the bounds, including odd-order groups with a cut-vertex and $h(G)=5$, thereby demonstrating the limits of the obtained results. Overall, the work connects the algebraic structure of solvable groups with the combinatorial geometry of $\Gamma(G)$ and provides sharp, realizable bounds.
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 $G$.