\begin{thebibliography}{13} %به جای عدد 8 تعداد کل مراجع قرار گیرد. \addcontentsline{toc}{chapterstar}{مراجع} \baselineskip=6mm \begin{LTRbibitems} \resetlatinfont \bibitem A K. Doerk and T. HAWKES, {\em Finite Soluble Groups}, Walter de Guyter, New York, 1992. \bibitem A B. HUPPERT, {\em Endliche Gruppen I}, Springer-Verlag, Berlin, 1967. \bibitem A I. M. ISAACS, {\em Character Theory of Finite Groups}, AMS Chelsea, Providence, 2006. (Corrected reprint of the 1976 original.) \bibitem A I. M. ISAACS. {\em Finite Group Theory}, Amer. Math. Soc., Providence, RI, 2008. \bibitem A A. YU. OLSHANSHKI, Groups of bounded period with subgroups of prime order, {\em Algebra i Logika} 21 (1982). 553--618. \bibitem [ AM17]A S. Aivazidis and T. Mu¨ller, On residuals of finite groups, arXiv:1708.04224 (2017). \bibitem [DH92]A K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter Expositions in Mathematics, vol. 4, Walter de Gruyter & Co., Berlin, 1992. \bibitem [GAP17]A The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.8, 2017. [Gas53] \bibitem A W. Gaschu¨tz, ¨Uber die Φ-Untergruppe endlicher Gruppen, Math. Z. 58 (1953), 160–170. \bibitem [HH56] A P. Hall and G. Higman, On the p-length of p-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42. \bibitem [Isa08]A I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008. \bibitem A †Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 9190401, Israel. \bibitem A ∗School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom %\bibitem{roe}W. Roelcke and S. Dierolf, Uniform Structures on Topological Groups and Their Quotients, \textit{McGraw-Hill International Book Co., New York}, 1981. \end{LTRbibitems} %\bibitem{dict} ا \end{thebibliography} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter*{فهرست نمادها} \addcontentsline{toc}{chapterstar}{فهرست نمادها} \markboth{فهرست نمادها}{فهرست نمادها} \vspace{-4.5mm} \begin{tabbing} \hspace{15.5mm}\=\parbox{\textwidth}{\hfill}\\ \notation{$C_G(A)$}:{ مرکزساز $A$ در $G$}{\pageref{notinteriora}} \notation{نرمالساز $H$ در $G$}:{$N_G(H)$}{\pageref{notclosurea}} \notation{}:{}{\pageref{notcardinal}} \notation{}:{}{\pageref{notdiagonal}} \notation{}:{}{\pageref{notdiscretemetric}} \notation{}:{}{\pageref{notfixt}} \notation{}:{}{\pageref{notgraph}} \notation{}:{}{\pageref{notidmapx}} \notation{}:{ }{\pageref{notnbhdsystemx}} \notation{}:{ }{\pageref{notnbhdxr}} \notation{}:{}{\pageref{notpowerset}} \notation{}:{}{\pageref{notplusreal}} \notation{}:{}{\pageref{notzeropseudometric}} \notation{}:{ }{\pageref{nottn}} \notation{}:{}{\pageref{notuinverse}} \notation{}:{ }{\pageref{notuvcomposite}} \notation{}:{ }{\pageref{notuniformnbhd}} \notation{}:{ }{\pageref{notsubset}} \notation{}:{}{\pageref{notpropersubset}} \notation{}:{}{\pageref{notplusinteger}} \end{tabbing} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ‌\chapter*{واژه‌نامه‌ی فارسی به انگلیسی} \addcontentsline{toc}{chapterstar}{واژه‌نامه‌ی فارسی به انگلیسی} \markboth{واژه‌نامه‌ی فارسی به انگلیسی}{واژه‌نامه‌ی فارسی به انگلیسی} \baselineskip=6mm \noindent{\textbf{\huge آ}}\\[1mm] {\textbf{\huge ا}}\\[1mm] \persiangloss{}{} \persiangloss{}{} \persiangloss{}{} \persiansubgloss{}{} \persiansubgloss{}{} \persiansubgloss{}{} \persiansubgloss{}{} \persiangloss{}{}\\ {\textbf{\huge ب}}\\[1mm] {\textbf{\huge پ}}\\[1mm] {\textbf{\huge ت}}\\[1mm] \persiangloss{تصویر همریخت}{Homomorphic image} \persiangloss{تکریختی}{Monomorphism} {\textbf{\huge ث}}\\[1mm] {\textbf{\huge ج}}\\[1mm] \persiangloss{حاصلضرب نیم مستقیم}{Semidirect product} \persiangloss{حاصلضرب حلقوی}{Wreath product} {\textbf{\huge چ}}\\[1mm] {\textbf{\huge ح}}\\[1mm] {\textbf{\huge خ}}\\[1mm] \persiangloss{خودریختی}{Automorphism} {\textbf{\huge د}}\\[1mm] \persiangloss{}{} \persiansubgloss{}{} \persiansubgloss{}{} \persiansubsubgloss{}{} \persiansubsubgloss{ }{}\\ {\textbf{\huge ذ}}\\[1mm] {\textbf{\huge ر}}\\[1mm] \persiangloss{رده پوچتوانی}{Nilpotency class} {\textbf{\huge ز}}\\[1mm] \persiangloss{زیرگروه نرمال آبلی ماکسیمال}{Maximal abelian normal subgroup} \persiangloss{زیرگروه بزرگ}{Large subgroup} \persiangloss{زیرگروه هال}{Hall subgroup} \persiangloss{زیرگروه فیتینگ}{Fitting subgroup} {\textbf{\huge ژ}}\\[1mm] {\textbf{\huge س}}\\[1mm] \persiangloss{سری نرمال}{Normal series} \persiangloss{سری زیرنرمال}{Subnormal series} \persiangloss{سری اصلی}{Main series} \persiangloss{سری آبلی}{Abel series} \persiangloss{سری مرکزی}{Central series} \persiangloss{سری مشتق}{Derivative series} \persiangloss{سرشت}{Character} \persiangloss{سرشت خطی}{Linear Character} \persiangloss{سرشت تحویل ناپذیر}{ Irreducible Character} \persiangloss{}{} \persiangloss{}{} {\textbf{\huge ش}}\\[1mm] {\textbf{\huge ص}}\\[1mm] {\textbf{\huge ض}}\\[1mm] {\textbf{\huge ط}}\\[1mm] {\textbf{\huge ظ}}\\[1mm] {\textbf{\huge ع}}\\[1mm] \persiangloss{عمل صادق}{Faithful action} \persiangloss{عمل اولیه}{Primitive action} \persiangloss{عمل متعدی}{Transitive action} {\textbf{\huge غ}}\\[1mm] {\textbf{\huge ف}}\\[1mm] {\textbf{\huge ق}}\\[1mm] {\textbf{\huge ک}}\\[1mm] {\textbf{\huge گ}}\\[1mm] \persiangloss{گروه حلپذیر}{Solvable group} \persiangloss{گروه فوق حلپذیر}{Supersolvable group} \persiangloss{گروه پوچتوان}{Nilpotent group} \persiangloss{گروه متناوب}{Alternating group} \persiangloss{گروه جایگشتی}{Permutation group} \persiangloss{گروه عامل}{Factor group} \persiangloss{گروه خطی عام}{General linear group} \persiangloss{گروه خطی خاص}{Special linear group} \persiangloss{گروه ساده}{Simple group} \persiangloss{}{} {\textbf{\huge ل}}\\[1mm] {\textbf{\huge م}}\\[1mm] \persiangloss{مرکزساز}{Centralizer} {\textbf{\huge ن}}\\[1mm] \persiangloss{نمایش جایگشتی}{Permutation representation} \persiangloss{نرمالساز}{Normalizer} \persiangloss{نمایش گروه}{Group representation} {\textbf{\huge و}}\\[1mm] {\textbf{\huge هـ}}\\[1mm] {\textbf{\huge ی}}\\[1mm] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter*{واژه‌نامه‌ی انگلیسی به فارسی} \addcontentsline{toc}{chapterstar}{واژه‌نامه‌ی انگلیسی به فارسی} \markboth{واژه‌نامه‌ی انگلیسی به فارسی}{واژه‌نامه‌ی انگلیسی به فارسی} \baselineskip=6mm \begin{latin} \noindent{\huge A}\\ \englishgloss{Automorphism}{خودریختی} \englishgloss{Alternating group}{گروه متناوب}\\ {\huge B}\\ \englishgloss{}{ } \englishgloss{}{} \englishgloss{}{} \englishgloss{}{}\\ {\huge C}\\ \englishgloss{Central series}{سری مرکزی} \englishgloss{Centralizer}{مرکزساز} \englishsubgloss{}{} \englishsubgloss{}{} \englishsubgloss{}{} \englishsubgloss{}{}\\ {\huge D}\\ \englishgloss{derived series}{سری مشتق} {\huge E}\\ \englishgloss{Endomorphism}{درونریختی} {\huge F}\\ \englishgloss{Factor group}{گروه عامل} \englishgloss{Faithful action}{عمل صادق} {\huge G}\\ \englishgloss{}{} {\huge H}\\ {\huge I}\\ \englishgloss{Image homomorphic}{تصویر همریخت} \englishgloss{ّIsomorphic}{یکریختی} {\huge J}\\ {\huge K}\\ {\huge L}\\ \englishgloss{Linear group}{گروه خطی} {\huge M}\\ {\huge N}\\ \englishgloss{Nilpotent group}{گروه پوچتوان} {\huge O}\\ {\huge P}\\ \englishgloss{Permutation group}{گروه جایگشتی} {\huge Q}\\ {\huge R}\\ {\huge S}\\ \englishgloss{Simple group}{گروه ساده} \englishgloss{Solvable group}{گروه حلپذیر} \englishgloss{Supersolvable group}{گروه فوق حلپذیر} \englishgloss{Special linear group}{گروه خطی خاص} {\huge T}\\ {\huge U}\\ {\huge V}\\ {\huge W}\\ \englishgloss{حاصلضرب حلقوی}{Wreath product} {\huge X}\\ {\huge Y}\\ {\huge Z}\\ \end{latin} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \chapter*{نمایه} \addcontentsline{toc}{chapterstar}{نمایه} \markboth{نمایه}{نمایه} \baselineskip=6mm \begin{multicols}{2} \noindent{\textbf{\huge آ}}\\[1mm] {\textbf{\huge ا}}\\[1mm] {\textbf{\huge ب}}\\[1mm] {\textbf{\huge پ}}\\[1mm] {\textbf{\huge ت}}\\[1mm] \termindex{}{\pageref{indcomposite}}\\ {\textbf{\huge ث}}\\[1mm] {\textbf{\huge ج}}\\[1mm] {\textbf{\huge چ}}\\[1mm] {\textbf{\huge ح}}\\[1mm] {\textbf{\huge خ}}\\[1mm] {\textbf{\huge د}}\\[1mm] {\textbf{\huge ذ}}\\[1mm] {\textbf{\huge ر}}\\[1mm] {\textbf{\huge ز}}\\[1mm] {\textbf{\huge ژ}}\\[1mm] {\textbf{\huge س}}\\[1mm] {\textbf{\huge ش}}\\[1mm] {\textbf{\huge ص}}\\[1mm] {\textbf{\huge ض}}\\[1mm] {\textbf{\huge ط}}\\[1mm] {\textbf{\huge ظ}}\\[1mm] {\textbf{\huge ع}}\\[1mm] {\textbf{\huge غ}}\\[1mm] {\textbf{\huge ف}}\\[1mm] {\textbf{\huge ق}}\\[1mm] \termindex{)}{\pageref{inddiagonal}}\\ {\textbf{\huge ک}}\\[1mm] {\textbf{\huge گ}}\\[1mm] {\textbf{\huge ل}}\\[1mm] {\textbf{\huge م}}\\[1mm] {\textbf{\huge ن}}\\[1mm] \termindexnocomma{} \termsubindex{}{\pageref{indfixedpoint}}\\ {\textbf{\huge و}}\\[1mm] \termindex{}{\pageref{indinverse}}\\ {\textbf{\huge هـ}}\\[1mm] \termindex{}{\pageref{indnbhd}}\\ {\textbf{\huge ی}}\\[1mm] \end{multicols} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \vspace*{0.5cm} \thispagestyle{fancy} \fancyhead[RO]{} \fancyhead[LO]{} \fancyfoot[C]{\lr{\thepage}} \renewcommand{\headrulewidth}{0pt} \renewcommand{\footrulewidth}{0pt} \begin{latin} \begin{center} \LARGE\textbf{Abstract} \end{center} \vhrulefill{1pt}\par In this thesis, we study the family of finite groups with the property that every maximal abelian normal subgroup is self-centralizing. It is well known that this family contains all finite supersolvable groups, but it also contains many other groups. In fact, every finite group $G$ is a subgroup of some member $\Gamma$ of this family, and we show that if $G$ is solvable, then $\Gamma$ can be chosen so that every abelian normal subgroup of $G$ is contained in some self-centralizing abelian normal subgroup of $\Gamma$. \par \vspace{-2mm}\noindent\vhrulefill{1pt}\\ \begin{description}[parsep=0mm,leftmargin=0cm] \item[Keywords:] Self-centralizing, Supersolvable residual, Wreath product, Abelian normal subgroup. \item[2010 Mathematics Subject Classification:] 20D10, 20D99 \end{description} \end{latin} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \vspace*{-2cm} \thispagestyle{empty} \baselineskip=8.5mm \begin{latin} \begin{figure}[ht] \centering \includegraphics[width=2.5cm]{kntuarm} \end{figure} \vspace{-1cm} \begin{center} \em{{\footnotesize K. N. Toosi University of Technology}\vspace{-3mm}\\ {\footnotesize Faculty of Mathematics}}\vspace{1.2cm}\\ {\LARGE Master Thesis}\vspace{-2mm}\\ {\normalsize In Partial Fulfillments for the Degree Master of Science in Pure Mathematics Algebra(Finite Group Theory)}\vspace{1.7cm}\\ \baselineskip=1cm \em{{\large Title:}}\vspace{-2mm}\\ {\LARGE\textbf{Large abelian normal subgroups}}\vspace{1.7cm}\\ \em{{\large By:}}\vspace{-2mm}\\ {\Large\textbf{Narges Kian}}\vspace{1.2cm}\\ \em{{\large Supervisor:}}\vspace{-2mm}\\ {\Large\textbf{Prof. A. R. Moghadamfar}}\vspace{1.2cm}\\ %{\large Supervisors:}\vspace{-2mm}\\ %{\Large\textbf{Dr. ...}}\vspace{-2mm}\\ %{\Large\textbf{Dr. ...}}\vspace{1.2cm}\\ %{\large Advisor:}\vspace{-2mm}\\ %{\Large\textbf{Dr. ...}}\vspace{1.7cm}\\ {\large September 2016} \end{center} \end{latin}