From 6fc5e0c7150c8c3d013a34058806467b320d3e53 Mon Sep 17 00:00:00 2001 From: Henry Haustein Date: Sat, 20 Oct 2018 12:55:36 +0200 Subject: [PATCH] GEO - endlich fertig, VL 19.10. --- .../Normalteiler_und_Quotientengruppen.tex | 115 +++++++++++++++++- 3. Semester/GEO/Vorlesung GEO.pdf | Bin 427528 -> 442725 bytes 2 files changed, 114 insertions(+), 1 deletion(-) diff --git a/3. Semester/GEO/TeX_files/Normalteiler_und_Quotientengruppen.tex b/3. Semester/GEO/TeX_files/Normalteiler_und_Quotientengruppen.tex index 65e25f0..32df7a8 100644 --- a/3. Semester/GEO/TeX_files/Normalteiler_und_Quotientengruppen.tex +++ b/3. Semester/GEO/TeX_files/Normalteiler_und_Quotientengruppen.tex @@ -70,4 +70,117 @@ Sei $G$ eine Gruppe. \item den $H\le G$ mit $N\le H$ und \item den $H\le\lnkset{G}{N}$ \end{itemize} -\end{lemma} \ No newline at end of file +\end{lemma} +\begin{proof} + \begin{itemize} + \item $\pi_N(H)=\{hN\mid h\in H\}=\{hnN\mid h\in H, n\in N\}=\lnkset{HN}{N}$ + \item Umkehrabbildung: $H\mapsto \pi_N^{-1}(H)$: \\ + $H\le\lnkset{G}{N}$: $\pi_N(\pi_N^{-1}(H))=H$, da $\pi_N$ surjektiv \\ + $N\le H\le G$: $\pi_N^{-1}(\pi_N(H))=\pi_N^{-1}(\lnkset{HN}{N})=HN\subseteq H\cdot H=H$ + \end{itemize} +\end{proof} + +\begin{proposition}[Homomorphiesatz] + \proplbl{1_3_8} + Sei $\phi:G\to H$ ein Gruppenhomomorphismus und $N\unlhd G$ mit $N\le \Ker(\phi)$. Dann gibt es genau einen Gruppenhomomorphismus $\overline{\phi}:\lnkset{G}{N}\to H$ mit $\overline{\phi}\circ\pi_N=\phi$. + \begin{center} + \begin{tikzpicture} + \node (V) at (0,0) {$G$}; + \node (W) at (3,0) {$H$}; + \node (R) at (1.5,-1.5) {$\lnkset{G}{N}$}; + \draw[->, above] (V) to node {$\phi$} (W); + \draw[->, left] (V) to node {$\pi_N$} (R); + \draw[->, right, dashed] (R) to node {$\overline{\phi}$} (W); + \end{tikzpicture} + \end{center} +\end{proposition} +\begin{proof} + Existiert so ein $\overline{\phi}$, so ist $\overline{\phi}(gN)=(\overline{\phi}\circ \pi_N)(g)=\phi(g)$ eindeutig bestimmt. Definiere $\overline{\phi}$ nun so. + \begin{itemize} + \item $\overline{\phi}$ ist wohldefiniert: $gN = g'N \overset{\propref{1_2_8}}{\Rightarrow}\exists g'=gn$ für ein $n\in N\Rightarrow \phi(g')=\phi(g)\cdot \underbrace{\phi(n)}_{=1}=\phi(g)$, da $n\in\Ker(\phi)$ + \item $\overline{\phi}$ ist Homomorphismus: $\overline{\phi}(gN\cdot g'N) = \overline{\phi}(gg'N)=\phi(gg')=\phi(g)\cdot\phi(g')=\overline{\phi}(gN)\cdot\overline{\phi}(g'N)$ + \end{itemize} +\end{proof} + +\begin{conclusion} + \proplbl{1_3_9} + Ein Gruppenhomomorphismus $\phi:G\to H$ liefert einen Isomorphismus + \begin{align} + \overline{\phi}:\lnkset{G}{\Ker(\phi)}\xrightarrow{\cong}\Image(\phi)\le H\notag + \end{align} +\end{conclusion} + +\begin{conclusion}[1. Homomorphiesatz] + Seien $H\le G$ und $N\unlhd G$. Der Homomorphismus + \begin{align} + \phi: H\overset{i}{\hookrightarrow} HN\xrightarrow{\pi_N}\lnkset{HN}{N}\notag + \end{align} + induziert einen Isomorphismus + \begin{align} + \overline{\phi}:\lnkset{H}{H\cap N}\xrightarrow{\cong}\lnkset{HN}{N}\notag + \end{align} +\end{conclusion} +\begin{proof} + \begin{itemize} + \item $\phi$ ist surjektiv: Für $h\in H$ und $n\in N$ ist + \begin{align} + hnN=hN=\phi(h)\in\phi(H)=\Image(\phi)\notag + \end{align} + \item $\Ker(\phi)=H\cap\Ker(\pi_N)=H\cap N$ + \end{itemize} + Mit \propref{1_3_9} folgt die Behauptung. +\end{proof} + +\begin{conclusion}[2. Homomorphiesatz] + Seien $N\unlhd G$ und $N\le H\unlhd G$. Der Homomorphismus $\pi_H:G\to\lnkset{G}{H}$ induziert einen Isomorphismus + \begin{align} + \lnkset{(\lnkset{G}{N})}{(\lnkset{H}{N})}\xrightarrow{\cong}\lnkset{G}{H}\notag + \end{align} +\end{conclusion} +\begin{proof} + Da $N\le H$ liefert $\pi_H$ einen Epimorphismus (mit \propref{1_3_8}) $\overline{\pi_H}:\lnkset{G}{N}\to \lnkset{G}{H}$. + \begin{center} + \begin{tikzpicture} + \node (V) at (0,0) {$G$}; + \node (W) at (3,0) {$\lnkset{G}{H}$}; + \node (R) at (1.5,-1.5) {$\lnkset{G}{N}$}; + \draw[->, above] (V) to node {$\pi_H$} (W); + \draw[->, left] (V) to node {$\pi_N$} (R); + \draw[->, right, dashed] (R) to node {$\overline{\pi_H}$} (W); + \end{tikzpicture} + \end{center} + Dieser hat Kern $\Ker(\overline{\pi_H})?\lnkset{H}{N}$, induziert nach \propref{1_3_9} einen Isomorphismus + \begin{align} + \lnkset{(\lnkset{G}{N})}{\Ker(\overline{\pi_H})}\xrightarrow{\cong}\Image(\overline{\pi_H})=\lnkset{G}{H}\notag + \end{align} +\end{proof} + +\begin{definition}[Konjugation] + Seien $x,x',g\in G$ und $H,H'\le G$. + \begin{enumerate}[label=(\alph*)] + \item $x^g:= g^{-1}xg$, \begriff{Konjugation} von $x$ mit $g$ + \item $x$ und $x'$ sind \begriff{konjugiert} (in $G$)$\Leftrightarrow\exists g\in G$: $x'=x^g$ + \item $H$ und $H'$ heißen \emph{konjugiert} (in $G$)$\Leftrightarrow\exists g\in G$: $H'=H^g=\{h^g\mid h\in H\}$ + \end{enumerate} +\end{definition} + +\begin{lemma} + Die Abbildung + \begin{align} + \Int:\begin{cases} + G\to \Aut(G) \\ g\mapsto (x\mapsto x^g) + \end{cases}\notag + \end{align} + ist ein Gruppenhomomorphismus. +\end{lemma} +\begin{proof} + \begin{itemize} + \item $\Int(g)\in\Hom(G,G)$: $(xy)^g=g^{-1}xyg=g^{-1}xgg^{-1}yg=x^g\cdot y^g$ + \item $(x^g)^h=h^{-1}g^{-1}xgh=(gh)^{-1}x(gh)=x^{gh}$ + \item $\Int(g)\in\Aut(G)$: Umkehrabbildung zu $\Int(g)$ ist $\Int(g^{-1})$ + \item $\Int(g)\in\Hom(G,\Aut(G))$: + \begin{align} + \Int(gh)=\Int(h)\circ\Int(g)=\Int(g)\cdot\Int(h)\notag + \end{align} + \end{itemize} +\end{proof} \ No newline at end of file diff --git a/3. Semester/GEO/Vorlesung GEO.pdf b/3. 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