On the partial \(\Pi\)-property of subgroups of finite groups. (Q2873311)

One of the most fruitful topics in the abstract theory of finite groups is the study of the influence of subgroup embedding properties on the structure of the group. In the paper under review, the authors consider the \textit{partial \(\Pi\)-property}: A subgroup \(H\) of \(G\) satisfies this property if there exists a chief series NEWLINE\[NEWLINE\Gamma_G\colon 1=G_0<G_1<\cdots<G_n=GNEWLINE\]NEWLINE of \(G\) such that for every \(G\)-chief factor \(G_i/G_{i-1}\) (\(1\leq i\leq n\)) of \(\Gamma_G\), NEWLINE\[NEWLINE|G/G_{i-1}:\text{N}_{G/G_{i-1}}(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})|NEWLINE\]NEWLINE is a \(\pi(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})\)-number. This property generalises a number of subgroup embedding properties, like partial CAP, \(\mathcal U\)-hypercentral embedding, S-quasinormality, X-permutability, S-embedding, \(\mathcal U\)-quasinormality,\dots Recall that a formation \(\mathcal F\) is a class of groups closed under taking epimorphic images and subdirect products, and that it is said to be solubly saturated if \(G\in\mathcal F\) whenever \(G/\Phi(N)\in\mathcal F\) for a soluble normal subgroup \(N\) of \(G\).NEWLINENEWLINE The main theorems obtained in this paper are the following:NEWLINENEWLINE Theorem A. Let \(\mathcal F\) be a solubly saturated formation containing all \(p\)-supersoluble groups and let \(E\) be a normal subgroup of \(G\) with \(G/E\in\mathcal F\). Let \(X\) be a normal subgroup of \(G\) such that \(\text F^*(E)\leq X\leq E\), where \(\text F^*(E)\) denotes the generalised Fitting subgroup of \(G\). Suppose that for any Sylow \(p\)-subgroup \(P\) of \(X\), every maximal subgroup of \(P\) satisfies the partial \(\Pi\)-property in \(G\). Then \(G\in\mathcal F\) or \(X/\text O_{p'}(X)\) is a quasisimple group with Sylow \(p\)-subgroups of order \(p\).NEWLINENEWLINE Theorem B. Let \(\mathcal F\) be a solubly saturated formation containing all \(p\)-supersoluble groups and let \(E\) be a normal subgroup of \(G\) with \(G/E\in\mathcal F\). Suppose that for any Sylow \(p\)-subgroup \(P\) of \(\text F^*(E)\), every cyclic subgroup of \(P\) of prime order or order \(4\) (when \(P\) is not quaternion-free) satisfies the partial \(\Pi\)-property in \(G\). Then \(G\in\mathcal F\).NEWLINENEWLINE Theorem C. Let \(\mathcal U\) be a solubly saturated formation containing all supersoluble groups and let \(E\) be a normal subgroup of \(G\) with \(G/E\in\mathcal U\). Let \(X\) be a normal subgroup of \(G\) such that \(\text F^*(E)\leq X\leq E\). Suppose that for any non-cyclic Sylow subgroup \(P\) of \(X\), either every maximal subgroup of \(P\) satisfies the partial \(\Pi\)-property in \(G\), or every cyclic subgroup of \(P\) of prime order or order \(4\) (when \(P\) is not quaternion-free) satisfies the partial \(\Pi\)-property in \(G\). Then \(G\in\mathcal F\).