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Keywords:
Gruenberg-Kegel graph; prime graph of finite group; cut-set; solvable group
Gruenberg-Kegel graph; prime graph of finite group; cut-set; solvable group
Summary:
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.
References:
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