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<p>Let Γ(<i>G</i>) be the Gruenberg-Kegel graph of a finite group <i>G</i>. We prove that if <i>G</i> is solvable and <i>σ</i> is a cut-set for Γ(<i>G</i>), then <i>G</i> has a <i>σ</i>-series of length 5 whose factors are controlled. As a consequence, we prove that if <i>G</i> is a solvable group and Γ(<i>G</i>) has a cut-vertex <i>p</i>, then the Fitting length ℓ<sub><i>F</i></sub>(<i>G</i>) of <i>G</i> 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 <i>G</i>, we give a geometrical description of Γ(<i>G</i>) when it has minimal cut-set of size 2.</p>

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Finite solvable groups whose Gruenberg-Kegel graph has a cut-set

  • 期刊论文

  • 发表日期：

    2025年11月21日

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  • Lorenzo Bonazzi

摘要

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 ℓF(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 Γ(G) when it has minimal cut-set of size 2.