Finite Solvable Groups in Which the -Quasinormality of Subgroups is a Transitive Relation

Abstract: Let <inline-formula id="IEq2"><alternatives><tex-math id="IEq2_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma=\{\sigma_{i} \mid i\in I\}$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>be a partition of the set of all primes, and let <inline-formula id="IEq3"><alternatives><tex-math id="IEq3_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula>be a finite group. The group <inline-formula id="IEq4"><alternatives><tex-math id="IEq4_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula>is said to be <inline-formula id="IEq5"><alternatives><tex-math id="IEq5_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq5.gif"></inline-graphic></alternatives></inline-formula>-primary if <inline-formula id="IEq6"><alternatives><tex-math id="IEq6_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq6.gif"></inline-graphic></alternatives></inline-formula>is a <inline-formula id="IEq7"><alternatives><tex-math id="IEq7_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{i}$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq7.gif"></inline-graphic></alternatives></inline-formula>-group for some <inline-formula id="IEq8"><alternatives><tex-math id="IEq8_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in I$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq8.gif"></inline-graphic></alternatives></inline-formula>and <inline-formula id="IEq9"><alternatives><tex-math id="IEq9_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq9.gif"></inline-graphic></alternatives></inline-formula>-complete if <inline-formula id="IEq10"><alternatives><tex-math id="IEq10_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq10.gif"></inline-graphic></alternatives></inline-formula>has a Hall <inline-formula id="IEq11"><alternatives><tex-math id="IEq11_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{i}$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq11.gif"></inline-graphic></alternatives></inline-formula>-subgroup for each <inline-formula id="IEq12"><alternatives><tex-math id="IEq12_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in I$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq12.gif"></inline-graphic></alternatives></inline-formula>. A subgroup <inline-formula id="IEq13"><alternatives><tex-math id="IEq13_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq13.gif"></inline-graphic></alternatives></inline-formula>of <inline-formula id="IEq14"><alternatives><tex-math id="IEq14_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq14.gif"></inline-graphic></alternatives></inline-formula>is (i) <inline-formula id="IEq15"><alternatives><tex-math id="IEq15_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq15.gif"></inline-graphic></alternatives></inline-formula>-subnormal in <inline-formula id="IEq16"><alternatives><tex-math id="IEq16_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq16.gif"></inline-graphic></alternatives></inline-formula>if it has a subgroup series <inline-formula id="IEq17"><alternatives><tex-math id="IEq17_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=A_{0} \leq A_{1} \leq \dotsb \leq A_{n}=G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq17.gif"></inline-graphic></alternatives></inline-formula>such that either <inline-formula id="IEq18"><alternatives><tex-math id="IEq18_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{i-1} \trianglelefteq A_{i}$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq18.gif"></inline-graphic></alternatives></inline-formula>or <inline-formula id="IEq19"><alternatives><tex-math id="IEq19_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{i}/(A_{i-1})_{A_{i}}$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq19.gif"></inline-graphic></alternatives></inline-formula>is <inline-formula id="IEq20"><alternatives><tex-math id="IEq20_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma}$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq20.gif"></inline-graphic></alternatives></inline-formula>-primary for each <inline-formula id="IEq21"><alternatives><tex-math id="IEq21_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1, \dots, n$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq21.gif"></inline-graphic></alternatives></inline-formula>; (ii) modular in <inline-formula id="IEq22"><alternatives><tex-math id="IEq22_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq22.gif"></inline-graphic></alternatives></inline-formula>if (1) <inline-formula id="IEq23"><alternatives><tex-math id="IEq23_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq23.gif"></inline-graphic></alternatives></inline-formula>for all <inline-formula id="IEq24"><alternatives><tex-math id="IEq24_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \leq G, Z \leq G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq24.gif"></inline-graphic></alternatives></inline-formula>such that <inline-formula id="IEq25"><alternatives><tex-math id="IEq25_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \leq Z$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq25.gif"></inline-graphic></alternatives></inline-formula>and (2) <inline-formula id="IEq26"><alternatives><tex-math id="IEq26_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq26.gif"></inline-graphic></alternatives></inline-formula>for all <inline-formula id="IEq27"><alternatives><tex-math id="IEq27_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y \leq G, Z \leq G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq27.gif"></inline-graphic></alternatives></inline-formula>such that <inline-formula id="IEq28"><alternatives><tex-math id="IEq28_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \leq Z$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq28.gif"></inline-graphic></alternatives></inline-formula>; (iii) <inline-formula id="IEq29"><alternatives><tex-math id="IEq29_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq29.gif"></inline-graphic></alternatives></inline-formula>-quasinormal in <inline-formula id="IEq30"><alternatives><tex-math id="IEq30_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq30.gif"></inline-graphic></alternatives></inline-formula>if <inline-formula id="IEq31"><alternatives><tex-math id="IEq31_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq31.gif"></inline-graphic></alternatives></inline-formula>is <inline-formula id="IEq32"><alternatives><tex-math id="IEq32_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq32.gif"></inline-graphic></alternatives></inline-formula>-subnormal and modular in <inline-formula id="IEq33"><alternatives><tex-math id="IEq33_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq33.gif"></inline-graphic></alternatives></inline-formula>. Finite solvable groups in which the <inline-formula id="IEq34"><alternatives><tex-math id="IEq34_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}</tex-math><inline-graphic href="11006_2023_2737_Article_IEq34.gif"></inline-graphic></alternatives></inline-formula>-quasinormality of subgroups is a transitive relation are described. Some known results are generalized.