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zNYHA48pyMbig1@%u#o4J24D>C=?zdSC~e#uf^#!hh=Fw!?js|B0k)aA|MhQp$u_*M euCSlKqpyFkuZycNU;>QK1cjF_UAxgx`2PSral7yU diff --git a/skript.tex b/skript.tex index 06d6339..8e85669 100644 --- a/skript.tex +++ b/skript.tex @@ -511,5 +511,118 @@ $p$ teilt $\abs{G : C_G(g_i)} \Rightarrow p $ teilt die Summe $\Rightarrow p $ t Rest: Übung. \end{bew} +\begin{bem} +Berühmte Ergebnisse: +\begin{enumerate}[I)] + \item Brunside's $pq$-Theorem: Seien $p, q$ Primzahlen, $\abs{G} = p^a \cdot q^b, a, b \in \N \Rightarrow G$ ist auflösbar. + \item Feit-Thompson: Ist $2 \nmid \abs{G} \Rightarrow G$ ist auflösbar. +\end{enumerate} +\end{bem} + +\underline{Beachte:} Sei $H \leq G$. Dann ist $H \trianglelefteq G \Leftrightarrow H$ ist Vereinigung von (disjunkten) Konjugationsklassen von $G$; denn +$gHg^{-1} = H \forall g \in G$ gilt genau dann, wenn $\forall h \in H, g \in G: c_g(h) = ghg^{-1} \in H$, d.h. $\lsup{G}{h} \subseteq H$. Daher ist +$\abs{H} = \sum\limits_{g_i \in H} \abs{C_i} $ % i ) 1, \ldots, k + +\underline{Erinnerung:} Sei $H \leq G$. Dann ist $N_g(H) = \set{g \in G \mid gHg^{-1} = H} \leq G$ und $H \trianglelefteq N_G(H) = $ die eindeutig bestimmte größte Untergruppe von $N$, +in der $H$ normal ist. $H \trianglelefteq G \Leftrightarrow N_G(H) = G$. + +\begin{satz} % 1.3.15 +Sei $\abs{G} = n < \infty$, und sei $H \leq G$. Sei $\mathcal{A} = \set{gHg^{-1} \mid g \in G}$. Dann ist $\abs{\mathcal{A}} = \abs{G : N_G(H)}$. +\end{satz} + +\begin{bew} +$G$ operiert auf $\sigma(G)$ ($\set{K \leq G}$) per Konjugation, und $\mathcal{A}$ ist gerade die Bahn $\lsup{G}{H}$ von $H$ unter dieser Operation. +$N_G(H) = \Stab_G(H)$. So folgt die Behauptung aus 1.3.7. +\end{bew} + +\begin{definition} +$H, K \leq G, z \in G$. Dann heißt $HzK = \set{hzk \mid h \in H, k \in K}$ die $H$-$K$-Doppelnebenklasse von $z$. \\ +Definiere $\sim$ auf $G$ durch $, y \in G$, so ist $x \sim y \Leftrightarrow \exists h \in H, k \in K: y = hxk$ +\begin{enumerate}[i)] + \item $ x = 1_H x 1_K \Rightarrow x \sim x \forall x \in G$ + \item $ y = hxk \Rightarrow x = h^{-1}yk^{-1} \Rightarrow$ Symmetrie + \item $ y = h_1 x k_1, z = h_2 y k_2 \Rightarrow z = h_2 h_1 x k_1 k_2 \Rightarrow x \sim z$ +\end{enumerate} +Also ist $G$ disjunkte Vereinigung der $H$-$K$-Doppelnebenklassen. +\end{definition} + +\underline{Klar:} $HzK = \bigcup\limits_{h \in H} hzK = \bigcup\limits_{k \in K} Hzk$ ist (disjunkte) Vereinigung von $K$-Links- bzw $H$-Rechtsnebenklassen in $G$. + +\begin{satz} % 1.3.16 +Sei $\abs{G} = n < \infty, H, K \leq G$ und $ \in G$. Dann gilt: \\ +$$ \abs{HzK} = \frac{\abs{H} \cdot \abs{K}}{\abs{H \cap zKz^{-1}}} = \frac{\abs{H} \cdot \abs{K}}{\abs{z^{-1}Hz \cap K}} = [H : (H \cap zKz^{-1}] \cdot \abs{K} = \abs{H} \cdot [K : (z^{-1}Hz \cap K] $$ +Das kommt nicht von unefähr: Ist $h_1 = 1, h2, \ldots,h_l \in H$ ein Vertretersystem der Linksnebenklassen von $H \cap zKz^{-1}$ in $H$, d.h. $H = \mathop{\cup}\limits^{\bullet} h_i (H \cap zKz^{-1}$, +so ist $HzK = \bigcup\limits_{j=1, \ldots, l}^{\bullet} h_i z K$. Analog $K = \bigcup\limits_{j=1, \ldots, m}^{\bullet} (z^{-1}Hz \cap K) \cdot k_j, HzK = \bigcup\limits_{j=1, \ldots, m}^{\bullet} Hz k_j $ +\end{satz} + +\underline{Beweisidee:} $h \in H \cap zKz^{-1} \Leftrightarrow \exists k \in K: h = zkz^{-1} \Leftrightarrow hzK = zkz^{-1}z K = zkK = zK \Rightarrow h_i (z^{-1}H \cap K) zK= h_i zK$, Details Übung. + +\begin{bew} +\begin{enumerate}[a)] + \item $\abs{HzK} = \abs{HzKz^{-1}} \mathop{=}\limits^{\text{1.3.12}} \frac{\abs{H} \cdot \abs{zKz^{-1}}}{\abs{H \cap zKz^{-1}}} = \frac{\abs{H} \cdot \abs{K}}{\abs{H \cap zKz^{-1}}} = \frac{\abs{H} \cdot \abs{K}}{\abs{z^{-1}Hz \cap K}}$ + \item 2. Beweis: $G$ operiert auf $G / K$ wie üblich, also auch $H$ durch Einschränkung. $HzK$ ist die Vereinigung der Nebenklassen, die in $\lsup{H}{zK}$ liegen. Daher ist $\abs{HzK} = \abs{K} \cdot $ Bahnlänge $\abs{\lsup{H}{zK}}$. Nun ist $\Stab_H(zK) = \set{h \in H \mid hzK = zK}$. Aber $hzK = zK \Leftrightarrow z^{-1}hz = k \in K \Leftrightarrow h = zkz^{-1}$ ist $\exists k \in K$. \\ + Also ist $\Stab_H(zK) = H \cap zKz^{-1} \mathop{\Rightarrow}\limits^{\text{1.3.7}} \abs{HzK} = \abs{K} [H : H \cap zKz^{-1}] = \frac{\abs{H}\cdot \abs{K}}{\abs{H \cap zKz^{-1}}}$ + \end{enumerate} +\end{bew} + +% \part{Lineare Gruppen} + +$F$ ist ein Körper, $n \in \N, G = \GL_n(F) \cong \Aut_F(V), v = F$-Vektorraum mit $\dim_F(V) = n$ \\ +$G = \set{A \in F^{n \times n} \mid \det A \neq 0} = $ volle lineare Gruppe. \\ +$SL_n(F) = \set{A \in F^{n \times n} \mid \det A = 1} = $ spezielle lineare Gruppe. \\ +$Z(G) = \set{\alpha \cdot E_{n \times n} | 0 \neq \alpha \in F}$ \\ +$Z(\SL_n(F)) = Z(G) \cap \SL_n(G)$, da $G = \SL_n(F) \cdot \set{\pmatr{XXX}}, Z(\SL_n(F)) = \set{ \alpha \cdot 1_G \mid 0 \neq \alpha \in F, \alpha^n = 1}$ \\ +$\PSL_n(F) = \SL_n(F) / Z(\SL_n(F)) = $ "`projektive spezielle lineare Gruppe"' \\ +\underline{Ziel:} $\Gamma = \PSL_n(F), \Gamma \neq \PSL_n(\GF(q))$ für $n = 2, q = 2, 3$ und $n = 3, q = 2$, dann ist $\PSL_n(F)$ einfach. + +\underline{Notation:} $\abs{F} = \GF(q) = \F_q$ Körper mit $q$ Elementen, $q = p^a, p$ Primzahl, $a \in N$. \\ +$G = \GL_n(q), \SL_n(F) = \SL_n(q), \PSL_n(F) = \PSL_n(q)$ \\ +$\abs{G} = \prod\limits_{k=1}^{n} (q^n - q^{k-1}) = (q^n-1)(q^n-q)(q^n-q^2)\ldots = q^{\frac{n(n-)}{2}} (q^n-1)(q^{n-1}-1)\ldots(q-1)$ +$\abs{\SL_n(q)} = \frac{\abs{\GL_n(q)}}{q-1}, \abs{\PSL_n(q)} = \frac{\abs{\SL_n(q)}}{\abs{ \alpha \mid \alpha^n = 1 \in F}} $ + +\chapter{Basics und ... Zerlegung} + +\begin{satz} % 2.1.1 +$\abs{\GL_n(q)} = $ oben (Algebra) +\end{satz} + +\begin{definition} +$T := \set{ \text{Diagonalmatrizen in } G = diag(\alpha_1, \ldots, \alpha_n) \mid 0 \neq \alpha_i \in \F_q}, \abs{T} = (q-1)^n$, Standard (Split) "`Torus"' \\ +$B := \set{ A = \text{obere Dreiecksmatrizen in } G \mid \det A = \prod \alpha_i \neq 0}$, Standard "`Boreluntergruppe"' von $G$ \\ +Klar: $T \leq B \leq G$ "`Borus"', "`Torel"'; $A \in B \Rightarrow A^{-1} \in B; X, Y \in B \Rightarrow XY \in B$, also $B \leq G$. \\ +$U = \set{ A \in B \mid \text{Diagonaleinträge von $A$ sind 1}} \leq B$ \\ +$A \in B, X \in U: AXA^{-1} \in U \Rightarrow U \trianglelefteq B$ +\end{definition} + +\underline{Klar:} $U \cap T = (1_G)$. \\ +Sei $A = \in B \Rightarrow A \cdot \pmatr{A_{11}^{-1} & & 0 \\ & \ddots & \\ & & A_{nn}^{-1}} \in U$ \\ +d.h. $Y \in T, A \cdot Y = X \Rightarrow A = X \cdot Y^{-1}$. Also ist $B = U \cdot T$. + +\begin{definition} +\begin{enumerate}[i)] + \item Eine Untergruppe von $G$, die konjugiert zu $B$ ist, heißt Boreluntergruppe von $G$. + \item Sei $\xi = (e_1, \ldots, e_n)$ natürliche Basis von $\F_q^n$, Für $\pi \in \sigma_n$ sei $\xi_{\pi} = (e_{\pi(1)}, \ldots, e_{\pi(n)})$. \\ + Sei $E_{\pi} = m_{\id}(\xi, \xi_\pi) $ Basiswechselmatrix von $\xi$ nach $\xi_\pi$ \\ + Beispiel: $\pi = (2,3,1): E_\pi = \pmatr{0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0} = $ Permutationsmatrix zu $\pi$. \\ + Beachte: Matrix-Einheit: $e_{ij} = (\delta_{r s}) \in M_{n \times n}(\F_q) $ \\ + $E_\pi = \sum\limits_{i=1}^n e_{\pi(i)i}$ +\end{enumerate} +\end{definition} + +\begin{definition} +Eine Permutationsmatrix $A \in M_{n \times n}(F)$ ist eine Matrix, die in jeder Spalte und Zeile genauen einen von 0 verschiedenen Eintrag hat, der 1 ist. \\ +Sei $A$ Permutationsmatrix. Definiere $\pi: \set{1, \ldots, n} \rightarrow \set{1, \ldots, n}$ durch $\pi(i) = j \Leftrightarrow A_{\pi(i)j} = 1$. +\end{definition} + +Also ist $\pi \mapsto E_\pi$ eine Bijektion von $\sigma_n$ in $W := \set{\text{Permutationsmatrizen}}$. + +Seien $\sigma, \pi \in \sigma_n$. Dann ist $E_\pi \cdot E_\sigma = (\sum_{i=1}^n e_{\pi(i)i}) (\sum_{j=1}^n e_{\sigma(j)j}) = \sigma_{i,j} e_{\pi(i)i} e_{\sigma(j)j} += \sum_{j=1}^{n} e_{\pi\sigma(j)j} = E_{\pi \sigma}$. \\ +Also ist $\pi \mapsto E_\pi$ ein Isomorphismus von $\sigma_n$ in $W$, insbesondere ist $W \leq G$, $W$ heißt "`Weylgruppe"' von $G$. + +\begin{satz} % 2.1.2 + 2.1.3 +Die Menge $W$ der Permutationsmatrizen in $G$ ist Untergruppe von $G$ und isomoprh zu $\sigma_n$ +\end{satz} + \end{document} diff --git a/standard.tex b/standard.tex index 282d83b..ab2c813 100644 --- a/standard.tex +++ b/standard.tex @@ -39,6 +39,7 @@ \providecommand{\C}[0]{\mathbb{C}} \providecommand{\N}[0]{\mathbb{N}} \providecommand{\Z}[0]{\mathbb{Z}} +\providecommand{\F}[0]{\mathbb{F}} \providecommand{\cO}[0]{\mathcal{O}} \providecommand{\cC}[0]{\mathcal{C}} @@ -80,6 +81,10 @@ \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\Out}{Out} \DeclareMathOperator{\Stab}{Stab} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\SL}{SL} +\DeclareMathOperator{\PSL}{PSL} +\DeclareMathOperator{\GF}{GF} %%