\begin{thebibliography}{13} %به جای عدد 8 تعداد کل مراجع قرار گیرد. \addcontentsline{toc}{chapterstar}{مراجع} \baselineskip=6mm \begin{LTRbibitems} \resetlatinfont \bibitem K. Doerk and T. HAWKES, {\em Finite Soluble Groups}, Walter de Guyter, New York, 1992. \bibitem B. HUPPERT, {\em Endliche Gruppen I}, Springer-Verlag, Berlin, 1967. \bibitem I. M. ISAACS, {\em Character Theory of Finite Groups}, AMS Chelsea, Providence, 2006. (Corrected reprint of the 1976 original.) \bibitem I. M. ISAACS. {\em Finite Group Theory}, Amer. Math. Soc., Providence, RI, 2008. \bibitem A. YU. OLSHANSHKI, Groups of bounded period with subgroups of prime order, {\em Algebra i Logika} 21 (1982). 553--618. \bibitem [ AM17] S. Aivazidis and T.Mu¨ller, On residuals of finite groups, arXiv:1708.04224 (2017). \bibitem [DH92] K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter Expositions in Mathematics, vol. 4, Walter de Gruyter \& Co., Berlin, 1992. \bibitem [GAP17] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.8, 2017. \bibitem [Gas53] W. Gaschu¨tz, ¨Uber die Φ-Untergruppe endlicher Gruppen, Math. Z. 58 (1953), 160–170. \bibitem [HH56] 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] I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008. \bibitem †Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 9190401, Israel. \bibitem ∗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}