From b0f15f866c1c68bc74fb92feff9333d9d809375d Mon Sep 17 00:00:00 2001 From: broadwaylamb Date: Sun, 19 Jun 2016 04:01:16 +0300 Subject: [PATCH] =?UTF-8?q?Corrected=20errata=20in=2010.1=E2=80=9310.4=20s?= =?UTF-8?q?ubsections?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- algebra.pdf | Bin 1296488 -> 1296494 bytes group-theory.tex | 13 +++++++------ program-4.pdf | Bin 46014 -> 46013 bytes program-4.tex | 2 +- 4 files changed, 8 insertions(+), 7 deletions(-) diff --git a/algebra.pdf b/algebra.pdf index d13c3fedcb105af18a0aba46d7e2461c2a51b08c..b26f5df159d48db28dcb8e4039ea55b053bc4684 100644 GIT binary patch delta 47079 zcmV(vK_myPG-WvcR;uxRF0|FV;tq?7|LOQau}d|e~^oG`K5yMV*)(uSXYMx_zeAA)0p>s zX7qD(OTozWPJ%bd;p?tpAd;y>Fr&Ub`1fZh5Y;o&%jyW=97XQ5E*{l)k4egtDYf^*g4_7EYg%gX*t(2Tl>Yj_m6gAEjyHo^@R0X3IFMGq=}8{?uPcd1299zWjXo?%{h4^u4z7du{A{ zMf@Hy{Wq5X^m_<@ABC4;3BJFgj*G%OlmF}X0P3`oY*=aeQw!yl$=}82_|HIk9XEmj zvQCh~Dui;puM?9%5-vp4T>PabmS2nS#ogtpw 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Множество левых смежных классов $G$ по $H$ по модулю подпространства (см. определение~\ref{def:quotient_space}); однако, отсутствие коммутативности приводит к тому, что необходимо рассматривать два варианта обобщения: условие $v_1-v_2\in U$ из -определения~\ref{def:quotient_space} мы заменяем на $v_1v_2^{-1}$ в -одном варианте и на $v_2^{-1}$ в другом. Если группа $G$ абелева, то +определения~\ref{def:quotient_space} мы заменяем на $v_1v_2^{-1}\in U$ в +одном варианте и на $v_2^{-1}v_1\in U$ в другом. Если группа $G$ абелева, то $gH = Hg$ для всех $g\in G$, и отношения $\sim_H$, ${}_H{\sim}$ совпадают. \end{remark} @@ -481,6 +481,7 @@ $ghg^{-1} = {}^gh$. абелевой группы нормальны. \end{remark} +\hspace{0em} \begin{examples}\label{examples:normal_subgroups} \hspace{1em} \begin{enumerate} @@ -573,7 +574,7 @@ $\ph(x)^{-1} = \ph(x^{-1})$. \begin{definition} Пусть $\ph\colon G\to H$~--- гомоморфизм групп. \dfn{Ядром} гомоморфизма $\ph$ называется множество $\Ker(\ph)=\{x\in G\mid -f\ph(x) = e_H\}$ (полный прообраз единицы). \dfn{Образом} гомоморфизма +\ph(x) = e_H\}$ (полный прообраз единицы). \dfn{Образом} гомоморфизма $\ph$ называется его теоретико-множественный образ: $\Img(\ph) = \{y\in H\mid y = \ph(x)\text{ для некоторого }x\in G\}$. \end{definition} @@ -621,14 +622,14 @@ $\ph$ сюръективно тогда и только тогда, когда $ \begin{lemma}\label{lem:injective_homo} Пусть $\ph\colon G\to H$~--- гомоморфизм групп. Он инъективен тогда и -только тогда, когда $\Ker(\ph) = e$. +только тогда, когда $\Ker(\ph) = \{e\}$. \end{lemma} \begin{proof} Если $\ph$ инъективен, то есть только один элемент $g\in G$ такой, что $\ph(g) =e$, и мы знаем, что $\ph(e)=e$. -Обратно, если $\Ker(\ph)=e$ и $g,g'\in G$ таковы, что +Обратно, если $\Ker(\ph)=\{e\}$ и $g,g'\in G$ таковы, что $\ph(g)=\ph(g')$, то $\ph(g^{-1}g') = \ph(g)^{-1}\ph(g') = e$, поэтому -$g^{-1}g'\in\Ker(\ph)=e$, откуда $g = g'$. +$g^{-1}g'\in\Ker(\ph)=\{e\}$, откуда $g = g'$. \end{proof} \begin{definition} diff --git a/program-4.pdf b/program-4.pdf index 1b581686bce9a18ad19cc30da6800c0cb0973ad5..0032a1e267ee18e34b61610d9024004cceb9e278 100644 GIT binary patch delta 658 zcmdn@oN4cKrU|Ny7Lx@xYi#rjWUbfS;pI>$^sr>f?MX$ijpMVqc(_dC!_yg?`={w)-oKZ}!b+ zp_%SaY@AOy-Z0y|A*tC}pgEOY($cu(``n7DUI8r^n64<-UdUZ(5W~Dn?&vJ1?1?J% zzZd2CP87P>lPdE5YF_r#%Z)FY9m1?uM6oV!=8@WeuhvpDDt>3=>^R#!Wx)%AP4l!p z@2W}f{`+EEt9j!NEw*)~=F_MC+T6fYbo201ZPz<&+v>Z8(idu~v&VO8hVC!-^U*S& zz4x|ZN7Tg4ug*s|-Lui68JI`@&{R(?H?jrA7oi}JR!h(7vOm3DgNsa>(A9KKWqC|x>7SC=7m*QtZ=t>Vx!wE{^Kiuv^w01 z+kPg9RZDiGsZy4sxCiS_#*eS!_GDjvv3mQuiPxVn`TlJe-xIfWa<57Ju2Zc#B5%)l zf8F~h!s3Xp*~1HNzrUR~NnV~(9L=9*mN)Hf`_wcsj;}2q(P!p{s~i7~cf2Eg;rO{Q z`Ou2}e}8>^D?d@NJ9x4qukhwH_NK+`mIj6<=Eeq-->;H^u{R5?UdQfaV(DyQYGPq% zY3X9@>S$!>WNcw>Xy{~WZeZ@{YGG=pU}H)^$z-O@PC<9KC~`Y;IPnB+5NYJlVQHAl zvU2~9=YJUwh$}8`;0TC*`D2dRcOma1H5+ZGDsbNA`o*~>eRujKZ^hmzK^r6AKDUSq iJD4rU|NymXifGYi#rjWUXKMNchR4T>&pv%U#vU-fsWdbUDMr%?vB`Opm9k z`Y+s}ce-q=sd>Dn)9mcxT#K^#0{5bSavyr@r084zX6J5?HM{aNmIbbwbz$qPwd$Es zr_J7bSt~u44&pnsZDp&dv`mwvj`gV%a~JOZeq^;oRLcdXD~oF{yj^J!!@f)I=q#`7 zjrA(O7v%*`6uQ{+ROJ2Dyxh5~mifqYY`Wz5Rm(y)<G;FTXC>FZ z;-(a)R$+OnLCzAMi&V4ul!r!Sl@NjQlxaZ>I#AF*Nvn7Lbpo$ z>G-(6k}Ft~HNRj=S?%%P#+@g3Usm^6S6}+gI`>`X{}bMdkDadXd^Ia_P8X|GrjyL_ zN8(JG)o+cwMOTy@xi(=%YpjCk6SX6&e}7yTyLkPBU2SobxlO*x$9<`~8fJE><)Yk4 zv97XbOMlhX>Bu}XUsjN@xc2w*SsRl#?cB<4ENy=J+hH$b?v|>cCA!bft)4vd-+P63 zf-fGYukP2{bHDEIk8k{jCwm1p2eD6G%x-LGZfI$2Jo()!85n!B;OcekPHwKo7S7Iw z#x9o522N(?&IZoTj^+mDuI6Trj+TxVb_zD81e8o>-0T$OyIqmnlf#K;Zqr;Imjp$I zGNC2^9?kpC&Jf?!#H8f#HuXXAO&c#Q&eIvydX7fBcyq)zM8A#m(41Paa_PeRZ=PG+ jzc%f6;@gx*wN3d)_{FujL?=((tiWN(rK;-c@5TiH?dT~O diff --git a/program-4.tex b/program-4.tex index 23ad4d8..40f227e 100644 --- a/program-4.tex +++ b/program-4.tex @@ -63,7 +63,7 @@ \glava{Теория групп} \resume{compactenum} \item Группы: определение, примеры. -\item Подгруппы: определение, примеры. Подгруппы циклической группы. +\item Подгруппы: определение, примеры. Подгруппы аддитивной группы. \item Подгруппа, порожденная множеством: две конструкции \item Классы смежности, разбиение на классы и соответствующие отношения эквивалентности.