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z&~^JjkL&#t;KPBt7Fwo3`$}h392|r*txMoQ$4KjUpv|S<44xUKl^BOW(^mChC0^*{ciG1-zC{ zV!#f~Q32M0Im5cm&|Xkvk-nXUi4s8z$F64S?;i~ARDt%v7eUmF{5<`^{{R={|LI9N zs-tz$Iw+(b9Ii`1?9u@Pap2fGy1L+S!RnxKdT33l|NjyvDNRlGAn2Lnus{z_c96C} O&SYhmmDM-j&HjJT@hueq diff --git a/skript.tex b/skript.tex index 0cbeb4e..e6d2bd7 100644 --- a/skript.tex +++ b/skript.tex @@ -1020,4 +1020,89 @@ Relativ leichte Übung: Ist $G$ einfach und $\abs{G} < 60$ so folgt $G \cong C_{ Beweis: Sei $g \in G$. Wegen $gNg^{-1} = N \trianglelefteq G$ ist $gPg{-1} \subseteq N \Rightarrow gPg^{-1} \in \Syl_p)N)$. Also gibt es $n \in N: n ( gPg^{-1} ) n^{-1} = P = ng P (ng)^{-1}$ $ \Rightarrow ng \in N_G(P) \Rightarrow g \in n^{-1} N_G(P) \subseteq N N_G(P) = N_G(P) N$. \qed +% Kapitel 4 +\chapter{Normalteilerstruktur} + +% Para 1 +\section{Satz von Jordan-Hölder} + +Sei im folgenden $G$ eine beliebige Gruppe. + +4.1.1 Definition: Sei $\Omega$ eine Menge, dann heißt $G$ Gruppe mit Operatorenbereich $\Omega$ (kurz $\Omega$-Gruppe), falls es eine externe binäre Verknüpfung $\Omega \times G \rightarrow G: (\alpha, g) \mapsto \alpha g \in G$ gibt mit $\alpha (g_1 g_2) = (\alpha g_1) (\alpha g_2) \forall g_1, g_2 \in G, \alpha \in \Omega$. \\ +\underline{Äquivalente Formulierung:} Es gibt eine Abbildung von $\Omega \rightarrow \set{\sigma: G \rightarrow G \mid \sigma \text{ ist Gruppenhom.}}$. + +Eine Untergruppe $H$ der $\Omega$-Gruppe $G$ heißt \underline{zulässig} ($\Omega$-Untergruppe, $H \leq_\Omega G$), falls $\alpha h \in H \forall h \in H, \alpha \in \Omega$, und sie heißt zulässiger Normalteiler +($\Omega$-Normalteiler, $H \trianglelefteq_\Omega G$), wenn $H$ zusätzlich Normalteiler von $G$ ist. + +Klar: Homomorphismen von $\Omega$-Gruppen: $F: G \rightarrow X$ Gruppenhomomorphismus mit $f(\alpha g) = \alpha f(g)$, $G, X$ $\Omega$-Gruppen, $g \in G, \alpha \in \Omega$ \\ +Es gelten Isosätze, Kerne von $\Omega$-Homomorphismen sind $\Omega$-Normalteiler, Bilder sind $\Omega$-Untergruppen. \\ +Eine $\Omega$-Gruppe heißt \underline{einfach}, falls sie keine nichttrivialen $\Omega$-Normalteiler hat. + +4.1.2 Beispiele: $G: \Omega$-Gruppe +\begin{enumerate}[i)] + \item $\Omega = \emptyset$: zulässigen Untergruppen = Untergruppe von $G$, zulässigen Normalteiler = Normalteiler von $G$. + \item $\Omega = \Inn(G)$ ($\Omega = G$ operiert durch Konjugation auf $G$): zulässigen Untergruppen = zulässigen Normalteiler = Normalteiler von $G$. + \item $\Omega = \Aut(G)$: zulässigen Untergruppen = char. Untergruppen von $G$. + \item $G = (R, +)$ = add. Gruppe eines Rings $R$ mit 1 (alle Untergruppen sind Normalteiler) \\ + $\Omega = R$ operiert auf $G$ per Multiplikation von links (rechts). Die $\Omega$-Untergruppen von $(R, +)$ sind genau die Linksideale (Rechtsideale) von $R$. + Links-Rechts-Operation: $\Omega \times G \times \Omega \rightarrow G: (\alpha, g, \beta) \mapsto \alpha g \beta$ mit $(\alpha g) \beta = \alpha (g \beta)$. \\ + Für $(R, +)$ mit $\Omega = R$ sind dann die zulässigen Untergruppen die Ideale von $R$. + \item $M = G = $ additive abelsche Gruppe mit $R$-Modul, $R = $ Ring $ \ni 1$, $M$ ist eine $R$-Gruppe unter Linksmultiplikation mit Elementen von $R$. \\ + Die zulässigen $R$-Untergruppen = zulässige $R$-Normalteiler = Untermoduln (analog für Rechtsmoduln). +\end{enumerate} + +Jetzt sei $\Omega$ eine Menge und $G$ eine $\Omega$-Gruppe. + +Definition: Eine endliche Folge $G = G_0 >_\Omega G_1 >_\Omega G_2 >_\Omega \ldots >_\Omega G_r = (1)$ von $\Omega$-Untergruppen heißt \underline{Kompositionsreihe} von $G$, +falls $G_{i+1} \trianglelefteq_\Omega G_i$ und $G_i/G_{i+1}$ ist einfache $\Omega$-Gruppe. + +Beispiel: $\sigma_5 \trianglerighteq A_5 \trianglerighteq (1)$ ist eine Kompositionsreihe mit "`Kompositionsfaktoren"' $ \sigma_5 / A_5 \cong C_2, A_5 = A_5/(1)$. + +Definition: $N$ ist maximale normale $\Omega$-Untergruppe von $G$, falls $G \neq N \trianglelefteq_\Omega G$ und kein $\Omega$-Normalteiler von $G$ echt zwischen $N$ und $G$ existiert $\Rightarrow G/N $ einfache $\Omega$-Gruppe. + +Beachte: Für $\Omega$-Gruppen gelten die 3 Isomorphiesätze, und daher der Korrespondenzsatz 1.1.11. + +4.1.3 Satz: Endliche Gruppen ($\Omega = \emptyset$) besitzen Kompositionsreihen. Beweis klar. + +4.1.4 Korrolar: Sei $G$ endliche Gruppe ($\Omega = \emptyset$), und sei $N \trianglelefteq G$. Dann besitzt $G$ eine Kompositionsreihe "`durch"' N, d.h. $N$ kommt als eine der Untergruppen $G_i$ vor. \\ +Beweis: Sei $N = N_0 > N_1 > N_2 > \ldots > N_k = (1)$ Kompositionsreihe von $N$ und $G/N = H_0 > H_1 > \ldots > H_r = (1)$ Kompositionsreihe von $G/N$, $G_i =$ volles Urbild von $H_i$ in $G/N$, also +$G_i = \set{g \in G \mid gN \in H_i}$. \\ +1.1.11 $\Rightarrow G = G_0 > G_1 > \ldots > G_{r-1} > N = N_0 > \ldots > N_k = (1)$ Kompositionsreihe von $G$ durch $N$. \\ + +4.1.5 Lemma: Sei $G$ beliebige $\Omega$-Gruppe mit einer Kompositionsreihe. Sei $N \trianglelefteq_\Omega G$, dann besitzt $N$ ebenfalls eine Kompositionsreihe. \\ +Beweis: Sei $G = G_0 > G_1 > \ldots > G_r = (1)$ Kompositionsreihe von $G$. Sei $N_i = N \cap G_i$. Dann ist $N = N_0 \geq N_1 \geq \ldots \geq N_r = (1)$ \\ +Dann ist $N_{i+1} = G_{i+1} \cap N \trianglelefteq N_i = G_i \cap N$ und $N_i/N_{i+1} = (N \cap G_i)/(N \cap G_{i+1}) = (N \cap G_i)/((N \cap G_i) \cap (G_{i+1})) +\cong$ (2. Isosatz) $((N \cap G_i)G_{i+1})/G_{i+1} \trianglelefteq G_i/G_{i+1}$ (Korrespondenzsatz) \\ +Also ist, da $G_i/G_{i+1}$ einfach ist, entweder $N_i/N_{i+1} = (1)$ (d.h. $N_i = N_{i+1}$), oder $N_i/N_{i+1} \cong G_i/G_{i+1}$ einfach. \\ +So erhalten wir eine Kompositionsreihe von $N$ durch Streichung der Wiederholungen in $N = N_0 \geq N_1 \geq \ldots \geq N_r = (1)$. \qed + +Definition: Eine Kette $G = G_0 > G_1 > \ldots > G_r = (1)$ heißt $\Omega$-Subnormalkette, falls $G_{i+1} \trianglelefteq_\Omega G$ ist, und $\Omega$-Normalkette, falls $G_i \trianglelefteq_\Omega G$ ist. + +Seien $G = G_0 > G_1 > \ldots > G_r = (1)$ und $G = H_0 > H_1 > \ldots > H_r = (1)$ zwei Subnormalketten derselben Länge $r$. Dann heißen diese äquivalent, fall es ein $\rho \in \sigma_r$ gibt mit $G_{i-1}/G_i \cong H_{\rho(i)-1}/H_{\rho(i)}$ für $1 \leq i \leq r$. \\ +Klar: Dies ist eine Äquivalenzrelation auf der Menge der Subnormalketten der Länge $r$ von $G$. + +4.1.6 Satz (Jordan-Hölder): Sei $G$ eine $\Omega$-Gruppe und besitze $G$ eine Kompositionsreihe. Dann haben je zwei Kompositionsreihen von $G$ dieselbe Länge und sind äquivalent. \\ +Konsequenz: In einer Kompositionsreihe einer $\Omega$-Gruppe (= einfache $\Omega$-Gruppen), sind die vorkommenden einfachen Kompositionsfaktoren mit ihren Multiplizitäten eindeutig bestimmt (aber nicht die Reihenfolge). + +Beweis: Seien $G = G_0 > G_1 > \ldots > G_r = (1)$ und $G = H_0 > H_1 > \ldots > H_s = (1)$ zwei Kompositionsreihen von $G$. \\ +Induktion über $r$: \\ +$r = 0$: $G = (1)$ trivial. \\ +$r = 1$: $G \trianglerighteq (1)$ ist Kompositionsreihe $\Rightarrow G$ ist einfach $\Rightarrow s = s, H_1 = (1)$. \\ +Sei $r > 1$ und die Behauptung bewiesen für alle $\Omega$-Gruppen mit einer Kompositionsreihe der Länge $< r$. \\ +Ist $G_1 = H_1$, so hat $G_1 = H_1$ die Kompositionsreihe $G_1 > G_2 > \ldots > G_r = (1)$ der Länge $r-1$ und $H_1 > H_2 > \ldots > H_s = (1)$, die nach Induktionsvoraussetzung äquivalent sind und $r-1 = s-1$, also $r = s$ und mit $G_/G_1 = H/H_1$ fertig. \\ +Sei also $G_1 \neq H_1$. Wegen $G_1 \trianglelefteq G_0 = G, H_1 \trianglelefteq H_0 = G$ ist $G_1 \lneq G_1 H_1 \trianglelefteq G$. Da $G/G_1$ einfach ist, ist also $G_1 H_1 = G$. \\ +Sei $K = G_1 \cap H_1$, dann ist $G/G_1 \cong H_1/K$ und $G/H_1 \cong G_1/K$. \\ +Also sind $G_1/K$ und $H_1/K$ einfache $\Omega$-Gruppen. \\ +Beachte $K \trianglelefteq G$, also besitzt $K$ eine Kompositionsreihe $K = K_0 > K_1 > \ldots > K_t = (1)$. \\ +$G_1 > K_0 > K_1 > \ldots > K_t = (1)$ ist Kompositionsreihe von $G$ der Länge $t+1$, die nach Induktionsvoraussetzung äquivalent zu $G_1 > G_2 > \ldots > G_r$ ist, und $t+1=r-1$, analog $t+1=s-1$, also $r = s$. \\ +Wegen $G/G_1 \cong H_1/K$ und $G/H_1 \cong G_1/K$ sind die ursprünglichen Kompositionsreihen äquivalent. + +Beispiele: +\begin{enumerate}[i)] + \item $\Omega = \emptyset$, Kompositionsreihen sind Subnormalketten $G = G_0 > G_1 > \ldots > G_r = (1)$ mit $G_{i+1} \trianglelefteq G_i$ und $G_i/G_{i+1}$ einfache Gruppe. + \item $M$ ist $R$-Module, $R = K$-Algebra, $\dim_K(M) < \infty \Rightarrow M $ hat Kompositionsreihe. + \item $G$ ist $\Omega$-Gruppe mit $\Omega = \Inn(G)$ ($\Aut(G)$?) +\end{enumerate} + + \end{document}