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author | michael-OST <75078383+michael-OST@users.noreply.github.com> | 2021-05-01 17:36:44 +0200 |
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committer | michael-OST <75078383+michael-OST@users.noreply.github.com> | 2021-05-01 17:36:44 +0200 |
commit | 3bd00062af2b2bbd04a82172eacb185119d09387 (patch) | |
tree | 7fd7ab0b94bf6bdd01e8cddcce598805e27675d9 /vorlesungen/slides/6/normalteiler | |
parent | Handout animation deleted (diff) | |
parent | Merge remote-tracking branch 'upstream/master' (diff) | |
download | SeminarMatrizen-3bd00062af2b2bbd04a82172eacb185119d09387.tar.gz SeminarMatrizen-3bd00062af2b2bbd04a82172eacb185119d09387.zip |
Merge branch 'master' into Steiner
Diffstat (limited to '')
-rw-r--r-- | vorlesungen/slides/6/normalteiler/konjugation.tex | 77 | ||||
-rw-r--r-- | vorlesungen/slides/6/normalteiler/normal.tex | 79 |
2 files changed, 156 insertions, 0 deletions
diff --git a/vorlesungen/slides/6/normalteiler/konjugation.tex b/vorlesungen/slides/6/normalteiler/konjugation.tex new file mode 100644 index 0000000..70ce01f --- /dev/null +++ b/vorlesungen/slides/6/normalteiler/konjugation.tex @@ -0,0 +1,77 @@ +% +% konjugation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Konjugation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{``Basiswechsel''} +In der Gruppe $\operatorname{GL}_n(\Bbbk)$ +\[ +A' = TAT^{-1} +\] +$T\in\operatorname{GL}_n(\Bbbk)$ +\\ +$A$ und $A'$ sind ``gleichwertig'' +\end{block} +\uncover<2->{% +\begin{block}{Definition} +$g_1,g_2\in G$ sind {\em konjugiert}, wenn es +$h\in G$ gibt mit +\[ +g_1 = hg_2h^{-1} +\] +\end{block}} +\uncover<3->{% +\begin{block}{Beispiel} +Konjugierte Elemente in $\operatorname{GL}_n(\Bbbk)$ haben die +gleiche Spur und Determinante +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Konjugationsklasse} +Die Konjugationsklasse von $g$ ist +\[ +\llbracket g\rrbracket += +\{h\in G\;|\; \text{$h$ konjugiert zu $g$}\} +\] +\end{block}} +\vspace{-7pt} +\uncover<5->{% +\begin{block}{Klassenzerlegung} +\begin{align*} +G +&= +\{e\} +\cup +\llbracket g_1\rrbracket +\cup +\llbracket g_2\rrbracket +\cup +\dots +\\ +&\uncover<6->{= +C_e\cup C_1 \cup C_2\cup\dots} +\end{align*} +\end{block}} +\vspace{-7pt} +\uncover<7->{% +\begin{block}{Klassenfunktionen} +Funktionen, die auf Konjugationsklassen konstant sind +\end{block}} +\uncover<8->{% +\begin{block}{Beispiele} +Spur, Determinante +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/6/normalteiler/normal.tex b/vorlesungen/slides/6/normalteiler/normal.tex new file mode 100644 index 0000000..42336b9 --- /dev/null +++ b/vorlesungen/slides/6/normalteiler/normal.tex @@ -0,0 +1,79 @@ +% +% normal.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Normalteiler} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Gegeben} +Eine Gruppe $G$ mit Untergruppe $N\subset G$ +\end{block} +\uncover<2->{% +\begin{block}{Bedingung} +Welche Eigenschaft muss $N$ zusätzlich haben, +damit +\[ +G/N += +\{ gN \;|\; g\in G\} +\] +eine Gruppe wird. + +\uncover<3->{Wähle Repräsentaten $g_1N=g_2N$} +\uncover<4->{% +\begin{align*} +g_1g_2N +&\uncover<5->{= +g_1g_2NN} +\uncover<6->{= +g_1g_2Ng_2^{-1}g_2N} +\\ +&\uncover<7->{= +g_1(g_2Ng_2^{-1})g_2N} +\\ +&\uncover<8->{\stackrel{?}{=} g_1Ng_2N} +\end{align*}} +\uncover<9->{Funktioniert nur wenn $g_2Ng_2^{-1}=N$ ist} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<10->{% +\begin{block}{Universelle Eigenschaft} +Ist $\varphi\colon G\to G'$ ein Homomorphismus mit $\varphi(N)=\{e\}$% +\uncover<11->{, dann gibt es einen Homomorphismus $G/N\to G'$:} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (N) at (-2.5,0); +\coordinate (G) at (0,0); +\coordinate (quotient) at (2.5,0); +\coordinate (Gprime) at (0,-2.5); +\coordinate (e) at (-2.5,-2.5); +\node at (N) {$N$}; +\node at (e) {$\{e\}$}; +\node at (G) {$G$}; +\node at (Gprime) {$G'$}; +\node at (quotient) {$G/N$}; +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (N) -- (G); +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (N) -- (e); +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (e) -- (Gprime); +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm] (G) -- (Gprime); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (G) -- (quotient); +\uncover<11->{ +\draw[->,shorten >= 0.3cm,shorten <= 0.4cm,color=red] (quotient) -- (Gprime); +\node[color=red] at ($0.5*(quotient)+0.5*(Gprime)$) [below right] {$\exists!$}; +} +\node at ($0.5*(quotient)$) [above] {$\pi$}; +\node at ($0.5*(Gprime)$) [left] {$\varphi$}; +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |