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\begin{document}
\frame{\maketitle}
\begin{frame}
\begin{lem}\cite{3}
	فرض می‌کنیم 
	$ \alpha $ عدد صحیح مثبتی باشد،
	به طوری که
	$ 2^\alpha - 1 $
	 یا
	$ 2^\alpha + 1 $
	یک عدد اول باشد. در این صورت 
	$ G \cong L_2(2^\alpha) $
	اگر و تنها اگر 
	\begin{itemize}
		\raggedright
		\item 
	$ |G| = |L_2(2^\alpha)| $
	\item
		$ b(G) = b(L_2(2^\alpha)) $،	
		که در آن 
		$ b(G) $
		بزرگترین درجه سرشت تحویل‌ناپذیر $ G $ را نشان می‌دهد.
	\end{itemize}
\raggedleft
\LTRfootnote{Jiang, Qinhui; Shao, Changguo. Recognition of $L_2(q)$ by its group order and largest irreducible character degree. Monatsh. Math. 176 (2015), no. 3, 413--422.}
\end{lem}

\end{frame}
\section{}
\begin{frame}{تشخیص ‌پذیری $L_3(q)$}
\textbf{سؤال:}
آیا ارتباطی بین کامل بودن گراف $ L_3(q) $ و تشخیص پذیر بودن آن هست؟
\begin{thm}
	فرض کنیم $ G \cong L_3(q) $ و
	$ q > 2 $
	توانی از یک عدد اول $ p $ باشد، در این صورت 
		گراف 
		$ \Gamma(G) $
		کامل است اگر وتنها اگر $ q $ فرد باشد و 
		$ q-1 = 2^i3^j $
		برای
		$ i \ge 1 $
		و
		$ j \ge 0 $. \LTRfootnote{Zhang, Runshi; Liu, Shitian. A characterization of linear groups $L_3(q)$ by their character degree graphs and orders. Bol. Soc. Mat. Mex. (3) 24 (2018), no. 1, 123--131.}
\end{thm}
\end{frame}
\section{ تشخیص پذیری گروههای $L_3(q)$}
\begin{frame}
\begin{thm}
	گروههای ساده متناهی $ L_3(q) $، که درآن 
	$ 2 \le q \le 4 $
	با استفاده از مرتبه و گراف درجه سرشت تشخیص پذیر هستند.
	\end{thm}
\pause
		توجه داریم که
		$ L_3(2) \cong L_2(7) $.
		
		\pause
		\begin{thm}
			فرض کنیم
			$ L \in \{L_3(5), L_3(7), L_3(8)\} $.
			
			اگر $ G $ یک گروه متناهی باشد به طوری که
			$ \Gamma(G) = \Gamma(L)$
			و
			$ |G| = |L| $
			آنگاه 
			$ G \cong L $.
		\end{thm}
\end{frame}
\section{}
\begin{frame}[allowframebreaks]{مراجع}
\setbeamertemplate{bibliography item}{\insertbiblabel}
\begin{thebibliography}{9}
	\resetlatinfont
	\begin{LTRbibitems}
		
		\bibitem{1}
		Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham, 1985.
		
		\bibitem{2}
		Isaacs, I. Martin. Character theory of finite groups. Corrected reprint of the 1976 original [Academic Press, New York]. Dover Publications, Inc., New York, 1994.
		
		\bibitem{3}
		Jiang, Qinhui; Shao, Changguo. Recognition of $L_2(q)$ by its group order and largest irreducible character degree. Monatsh. Math. 176 (2015), no. 3, 413--422.
		
		\bibitem{4}
		Khosravi, Behrooz; Khosravi, Behnam; Khosravi, Bahman. Recognition of $\text{PSL}(2,p)$ by order and some information on its character degrees where $p$ is a prime. Monatsh. Math. 175 (2014), no. 2, 277--282.
		
		\bibitem{5}
		Khosravi, Behrooz; Khosravi, Behnam; Khosravi, Bahman; Momen, Zahra. A new characterization for the simple group ${\rm PSL}(2,p^2)$ by order and some character degrees. Czechoslovak Math. J. 65(140) (2015), no. 1, 271--280. 
		
		\bibitem{6}
		Khosravi, Behrooz; Khosravi, Behnam; Khosravi, Bahman; Momen, Zahra. Recognition of some simple groups by character degree graph and order. Math. Rep. (Bucur.) 18(68) (2016), no. 1, 51--61.
		
		\bibitem{07}
		Khosravi, Behrooz; Khosravi, Behnam; Khosravi, Bahman; Momen, Zahra. Recognition of the simple group ${\rm PSL}(2,p^2)$ by character degree graph and order. Monatsh. Math. 178 (2015), no. 2, 251--257.
					
		\bibitem{7}
		Manz, Olaf; Wolf, Thomas R. Representations of solvable groups. London Mathematical Society Lecture Note Series, 185. Cambridge University Press, Cambridge, 1993.
		
		\bibitem{8}
		White, Donald L. Degree graphs of simple linear and unitary groups. Comm. Algebra 34 (2006), no. 8, 2907--2921.
		
		\bibitem{9}
		Zhang, Runshi; Liu, Shitian. A characterization of linear groups $L_3(q)$ by their character degree graphs and orders. Bol. Soc. Mat. Mex. (3) 24 (2018), no. 1, 123--131.
		
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	\raggedleft
%	\begin{persian}
%	
%	
%	\bibitem{ج}
%	جمالی، علیرضا؛ مباحثی در نظریۀ گروهها؛ انتشارات مبتکران؛ (1380).
%	
%	\bibitem{د.گ}
%	درفشه، محمدرضا؛ گروههای خطی؛ موسسه انتشارات و چاپ دانشگاه تهران؛ (1377).
%	
%	\bibitem{د.ن}
%	درفشه، محمدرضا؛ نمایش و سرشت گروه؛ مرکز نشر دانشگاهی تهران؛ (1381).
%	\end{persian}
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\end{frame}
%\section{مراجع}
%\begin{frame} {Refrences}
%
%\begin{itemize}
%\latin
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%	\raggedright
%	\item[{\rm [1]]  M. Akbari and A. R. Moghaddamfar, Groups for which the noncommuting graph is a split graph, {\em International Journal of Group Theory}, 6(1)(2017), 29--35.
%	\item[{\rm [2]}] X. Y. Chen, A. R. Moghaddamfar and M. Zohourattar, Some properties of various graphs associated with finite groups, Submitted.
%	\item[{\rm [3]}] S. F$\rm \ddot{o}$ldes and P. L. Hammer, Split graphs,
%	Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing.
%	\item[{\rm [4]}]  M. L. Lewis, D. V. Lytkina, V. D. Mazurov, A. R. Moghaddamfar, Splitting via noncommutativity, {\em Taiwanese Journal of Mathematics}, 2018.
%\persian
%
%\item
%معظمی، دارا؛ نظریه گراف و کاربردهای آن؛ مرکز نشر دانشگاهی تهران؛ (1378).
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