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On weakly s-semipermutable subgroups of finite groups. (English) Zbl 1269.20020
Summary: Suppose that \(G\) is a finite group and \(H\) is a subgroup of \(G\). We say that \(H\) is s-semipermutable in \(G\) if \(HG_p=G_pH\) for any Sylow \(p\)-subgroup \(G_p\) of \(G\) with \((p,|H|)=1\); \(H\) is weakly s-semipermutable in \(G\) if there are a subnormal subgroup \(T\) of \(G\) and an s-semipermutable subgroup \(H_{ssG}\) in \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_{ssG}\). The structure of a finite group with some weakly s-semipermutable subgroups is investigated. Mainly, we get the following local version’s result which is a uniform extension of many recent results in literature:
Main Theorem. Assume that \(p\) is a fixed prime in \(\pi(G)\) and \(E\) is a normal subgroup of \(G\) and \(Z_{\mathcal U\phi}(G)\) denotes the product of all normal subgroups \(H\) of \(G\) such that all non-Frattini \(p\)-\(G\)-chief factors of \(H\) have order \(p\). Then \(E\leq Z_{\mathcal U_p\phi}(G)\) if there exists a normal subgroup \(X\) of \(G\) such that \(F^*_p(E)\leq X\leq E\), where \(F^*_p(E)\) is the generalized \(p\)-Fitting subgroup of \(E\), and \(X\) satisfies the following: for any Sylow \(p\)-subgroup \(P\) of \(X\), \(P\) has a subgroup \(D\) such that \(1<|D|<|P|\) and all subgroups \(H\) of \(P\) with order \(|H|=|D|\) and all cyclic subgroups of \(P\) with order 4 (if \(P\) is a non-Abelian 2-group and \(|D|=2\)) are weakly s-semipermutable in \(G\).

MSC:
20D40 Products of subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D35 Subnormal subgroups of abstract finite groups
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