Criteria for the solubility and non-simplicity of finite groups.
Let \(G\) be a finite group. The set of all prime divisors of \(|G|\) is denoted by \(\pi(G)\). A Sylow set \(\Sigma(G)\) of \(G\) is a set of Sylow \(p\) subgroups of \(G\), one for each \(p\in\pi(G)\). \textit{S. Abe} and \textit{N. Iiyori} [Hokkaido Math. 29, No. 2, 391-407 (2000; Zbl 0965.20008)] introduced a soluble graph \(\Gamma_{sol}(G)\) which generalizes the well-known prime graph \(\Gamma(G)\). The vertex set of \(\Gamma_{sol}(G)\) is \(\pi(G)\) with \(\{p,q\}\) an edge if and only if \(G\) contains a soluble subgroup whose order is divisible by \(pq\). \textit{N. Iiyori} [in Commun. Algebra 30, No. 4, 1679-1691 (2002; Zbl 0997.20022)] determined all non-Abelian finite simple groups \(G\) for which the subgraph of \(\Gamma_{sol}(G)\) on the vertices \(\pi(G)\setminus\{2\}\) is complete. Generalizing this result, the authors prove the following Theorem 1.1: If \(G\) is a non-Abelian finite simple group and the subgraph of \(\Gamma_{sol}(G)\) on the vertices \(\pi(G)\setminus\{2,3\}\) is complete, then \(G\) is isomorphic to one of following groups: \(L_2(9)\), \(L_2(q)\) for some prime power \(q\) with \(\pi(q+1)\subseteq\{2,3\}\), \(L_3(3)\), \(U_3(3)\), \(U_4(2)\), \(M_{11}\), \(M_{12}\). As an immediate consequence of Theorem 1.1 the following Theorem 1.2 is obtained: If \(\Sigma(G)\) is a Sylow set of \(G\) such that for every pair \(P,Q\in\Sigma(G)\) the subgroup \(H=\langle x,y\rangle\) is soluble for each \(x\in P\), \(y\in Q\), then \(G\) is soluble; moreover, if the subgroup \(H=\langle x,y\rangle\) is supersoluble for each \(x\in P\), \(y\in Q\), then \(G\) has a Sylow tower. Further, the authors introduce the center \(Z(\Gamma_{sol}(G))\) of the graph \(\Gamma_{sol}(G)\) as the set of vertices \(p\in\pi(G)\) such that \(p\) is joined in \(\Gamma_{sol}(G)\) with every vertex \(s\in\pi(G)\), \(s\not=p\), and obtain the following non-simplicity criterion (Theorem 1.3): If \(Z(\Gamma_{sol}(G))\) contains the largest element of \(\pi(G)\), then \(G\) is no non-Abelian simple group. As some applications of Theorem 1.3, the following two theorems are proved. Theorem 1.4: If \(p\) is the largest element of \(\pi(G)\) and for every subgroup \(T\) of \(G\) of order \(p\) there exists a Sylow set \(\Sigma(G,T)\) of \(G\) such that \(PT=TP\) for each \(P\in\Sigma(G,T)\), then the minimal normal subgroups of \(G\) are \(p\)-soluble. Theorem 1.5: If \(G=AB\) where \(A\) and \(B\) are subgroups of \(G\) such that the largest element of \(\pi(G)\) belongs to \(Z(\Gamma_{sol}(A))\cap Z(\Gamma_{sol}(B))\), then \(G\) is no non-Abelian finite simple group.
scientific article; zbMATH DE number 2153333
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