On weakly \({\mathfrak{Z}}\) -permutable subgroups of finite groups II

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• 作者：A. A. Heliel ; M. M. Al-Shomrani ; T. M. Al-Gafri
• 关键词：20D10 ; 20D15 ; 20D20 ; 20F16
• 刊名：Arabian Journal of Mathematics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：5
• 期：1
• 页码：63-68
• 全文大小：769 KB
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• 作者单位：A. A. Heliel (1) (2)
  M. M. Al-Shomrani (2)
  T. M. Al-Gafri (2)
  
  1. Department of Mathematics, Faculty of Science 62511, Beni-Suef University, Beni-Suef, Egypt
  2. Department of Mathematics, Faculty of Science 80203, King Abdulaziz University, 21589, Jeddah, Saudi Arabia
  
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：2193-5351

文摘

Let G be a finite group. We say that \({\mathfrak{Z}}\) is a complete set of Sylow subgroups of G if for each prime p dividing the order of \({G, \mathfrak{Z}}\) contains exactly one Sylow p-subgroup of G, G _p say. A subgroup of G is said to be \({\mathfrak{Z}}\)-permutable in G if it permutes with every member of \({\mathfrak{Z}}\). A subgroup H of G is said to be weakly \({\mathfrak{Z}}\)-permutable in G if there exists a subnormal subgroup K of G such that G = HK and \({H \cap K \leq H_\mathfrak{Z}}\), where \({H_{\mathfrak{Z}}}\) is the subgroup of H generated by all those subgroups of H which are \({\mathfrak{Z}}\)-permutable in G . In this paper, we prove that G is supersolvable if the maximal subgroups of \({G_{p} \cap F ^{\ast}(G)}\) are weakly \({\mathfrak{Z}}\)-permutable in G, for every \({G_{p} \in \mathfrak{Z}}\), where \({F^{\ast} (G)}\) is the generalized Fitting subgroup of G. Also, we prove that if \({\mathfrak{F}}\) is a saturated formation containing the class of all supersolvable groups, then \({G \in \mathfrak{F}}\) if and only if there is a normal subgroup H in G such that \({G/H \in \mathfrak{F}}\) and the maximal subgroups of \({G_{p} \cap F^{\ast}(H)}\) are weakly \({\mathfrak{Z}}\)-permutable in G, for every \({G_{p} \in \mathfrak{Z}}\). Mathematics Subject Classification 20D10 20D15 20D20 20F16