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Sei $\epsilon = \det D / \det C = \det (DC^{-1}), A = \pmatr{\epsilon & 0 \\ 0 & 1_{n-1}}, \det A = \epsilon, B = DA^{-1}C^{-1}$, so ist $Bc_1 = \epsilon^{-1}d_1, Bc_i = d_i $ für $i > 2$ \\ $BFc_i = FBc_i = Fd_i $ für alle $i$. Klar: $\det B = 1$, d.h. $B \in \SL_n(F)$ \qed. +Missing: 27.11.2009 \\ +$ \abs{G} = p^a m, p \nmid m, \abs{G}_p = p^a, \abs{G}_{p'} = m $ + +$ \Syl_p(G) \equiv 1 \mod p $ +Beweis: $X = \set{A \subseteq G \mid \abs{A} = \abs{G}_p = p^a }$ $G$-Menge \\ +Bemerkung: $P \in \Syl_p(G) \Rightarrow P \in X$ \\ +$ A \in X$ so gibt es ein $g \in G: g \cdot A \ni 1$ \\ +$ \abs{X} = \pmatr{p^a \cdot m \\ p^a} = \sum_{O \in \text{$G$-Bahnen von $X$}} \abs{O} $ \\ +Sei $A \in X, A \in O = $ Orbit von $X$so, dass $1 \in A$. Sei $P = \Stab_G(A) \leq G$. Dann ist $P \subseteq P \cdot A = A \Rightarrow \abs{P} \leq \abs{A} = p^a $ \\ +1.3.7 $\Rightarrow \abs{O} = \abs{G:P}$. \\ +Angenommen $p$ teilt nicht $\abs{G:P}$, so ist $\abs{G}_p$ Teiler von $\abs{P}$. Also ist $\abs{G}_p = \abs{P} = p^a$ und $P \in \Syl_p(G)$ und $\abs{O} = m$. \\ +Sei umgekehrt $P \in \Syl_p(G)$. Dann ist die $G$-Menge $G/P = \cup g_i P$ mit $\abs{g_i P} = \abs{P} = p^a$, d.h. $G/P$ ist ein Orbit $O$ in $X$: $\abs{O} = \abs{G:P} = m$ \\ +Klar: $\Stab_G(1 \cdot P) = P$ \\ +Auf diese Weise erhalten wir eine Bijektion zwischen der Menge der $G$-Bahnen in $X' := \set{A \in X \mid p \nmid \abs{G \cdot A}}$ \\ +Also ist $X' = $ Vereinigung aller Bahnen $O$ von $X$ mit $p \nmid \abs{O}$ \\ +$X \without X' = $ Vereinigung aller Bahnen $O$ von $X$ mit $p \mid \abs{0}$ und daher $p \mid \abs{X \without X'} = \abs{X} - \abs{X'} $ \\ +Also $\abs{X} \equiv \abs{X'} \mod p$ \\ +Sei $r = \abs{\Syl_p(G)} = $ Anzahl der $p$-Sylow Untergruppen von $G$ = $\abs{\set{\text{Bahnen $O$ von $X$ mit $O \subseteq X'$}}}$. \\ +Es gilt dann: $r \cdot m = \abs{X'} \equiv_p \abs{X} \equiv_p \pmatr{p^a m \\ p^a}$ \\ +$ p \nmid m \Rightarrow r \mod p$ ist nur von $\abs{G}$ und nicht von $G$ selbst abhängig. Das heißt je zwei Gruppen $G$ und $H$ mit $\abs{G} = \abs{H}$ haben $\mod p$ dieselbe Anzahl von $p$-Sylowgruppen. \\ +Sei $G = C_{p^a m}$, dann ist $r \equiv 1 \mod p$, also ist $\abs{\Syl_p(G)} \equiv 1 \mod p$ für alle $G$ mit $G = p^a m$. Insbesondere ist $r \neq 0$, d.h. $G$ besitzt mindestens eine $p$-Sylow Untergruppe. \\ +Dies zeigt 1) und 4). + +Sei nun $P \in \Syl_p(G), G$ Gruppe der Ordnung $p^a m, p^a = \abs{G}_p, Q \leq G $ $p$-Gruppe. \\ +$Y = \set{gPg^{-1} \mid g\in G}$. $Q$ operiert auf $Y$ durch Konjugation: \\ +$\lsup{x}{(yPy^{-1})} = xyP (xy)^{-1} \in Y$ für $x \in X$. + +Sei $O$ ein $Q$-Orbit von $Y$, $P_1 \in O$. Dann ist $\abs{O} = \abs{Q : \Stab_Q(P_1)} = $ Potenz von p (möglicherweise $p^0$). \\ +Aber $\abs{Y} = \abs{G : N_G(P)}$ Teiler von $m$ (1.3.15) $\Rightarrow p \nmid \abs{Y}$.Also muss es eine $Q$-Bahn $O$ in $Y$ geben mit $p \nmid \abs{O} \Rightarrow \exists Q$-Bahn $O$ in $Y$ mit $\abs{O} = p^0 = 1 \Rightarrow O = \set{P_1}$. Dann ist also $xP_1 x^{-1} = P_1 \forall x \in Q$. Daher ist $Q P_1 = P_1 Q$ und mit (1.1.4) ist $Q P_1 \leq G$ \\ +Klar ist: $\abs{P_1} \leq \abs{Q P_1}$. Nach 1.3.12 ist $\abs{Q P_1} = \frac{\abs{Q}\abs{P_1}}{\abs{Q \cap p_1}} = \abs{P_1} \abs{Q: Q\cap P_1}$. Also ist $QP_1$ eine $p$-Untergruppe von $G$. \\ +Also ist wegen $\abs{P_1} \leq \abs{QP_1}$ die Ordnung $\abs{Q \cdot P_1}$ von $QP_1$ gleich $P_1 = p^a$ und daher $\abs{Q \cap P_1} = \abs{Q} \Rightarrow Q \subseteq P_1 \in \Syl_p(G)$. Dies zeigt 3). + +Seien $Q, P \in \Syl_p(G) \Rightarrow $ (nach vorigem Schritt) $\exists g \in G: Q \leq gPg^{-1}$. Wegen $\abs{Q} = \abs{P} = p^a$ ist $Q = gPg^{-1}$. \qed + +2. Beweis für Existenz von $p$-Sylowuntergruppen: + +Induktion über $\abs{G}$ \\ +$\abs{G} = 1$ trivial \\ +$p \nmid \abs{G}$ trivial \\ +$\abs{G} = p^a m$ mit $p \nmid m > 1$, und sei die Behauptung beweisen für alle Gruppen $\abs{H}$ mit $\abs{H} < \abs{G}$. \\ +Besitzt $G$ eie echte Untergruppe $H$ mit $p \nmid [G : H]$, so ist jede $p$-Sylowgruppe von $H$ eine $p$-Sylowgruppe von $G$ und wir sind fertig. \\ +Ohne Einschränkung gelte $H \lneq G \Rightarrow p \mid [G : H]$ \\ +Klassengleichung 1.3.13: \\ +$\abs{G} = \abs{Z(G)} + \sum\limits_{i=1}^l [G : C_G(g_i)]$ mit $\set{g_1, \ldots, g_l}$ Repräsentanten von Konjugationsklassen von $G$ der Größe $> 1$. \\ +Für $1 \leq i \leq l$ ist $C_G(g_i) \lneq G$ und daher $p \mid [G : C_G(g_i)]$ \\ +Also teilt $p \mid Z(G) =$ abelsche Gruppe. Also ist $\abs{Z(G)} > 1$. \\ +$\Rightarrow G$ besitzt eine normale Untergruppe $N$ ($\leq Z(G)$) der Ordnung $p$. $\abs{G/N} = p^{a-1}m < p^am \Rightarrow \exists \overline{P} \in \Syl_p(G/N)$ \\ +Sei $P = $ volles Urbild von $\overline{P}$ in $G \Rightarrow N \trianglelefteq P, P/N = \overline{P} \Rightarrow \abs{P} = p^a \Rightarrow P \in \Syl_p(G)$. + +Korrolar: Sei $\abs{G} = p^a m, p^a = \abs{G}_p$. Dann gibt es für $1 \leq b \leq a$ eine Untergruppe $H$ von $G$ mit $\abs{H} = p^b$ (Weil $P \in \Syl_p(G)$ eine $p$-Gruppe, daher nilpotent und damit auflösbar ist. Wir können $H \leq P$ wählen!). + +3.1.3 Korrolar: $\abs{\Syl_p(G)}$ ist Teiler von $\abs{G}_{p'} = \frac{\abs{G}}{\abs{G}_p}$ ($\abs{G} = p^a m, p \nmid m$) \\ +Beweis: $G$ operiert auf $\Syl_p(G)$ per Konjugation transitiv. Also ist $P \in \Syl_p(G)$, so ist $\abs{\Syl_p(G)} = [G : \Stab_G(P)]$. Wegen $P \trianglelefteq N_G(P) \leq G$ ist daher $\abs{\Syl_p(G)}$ Teiler von $m$. + +3.1.4 Korrolar: (Cauchy's Theorem) $G$ hat ein Element der Ordnung $p$ ($p \mid \abs{G}$) \\ +3.1.5 Satz: Sei $N \trianglelefteq G (N \neq G)$, und sei $P \in \Syl_p(G)$. Dann ist $PN/N \in \Syl_p(G/N)$ und $P \cap N \in \Syl_p(N)$. \\ +Beweis: $[G/N : PN/N] = [G : PN]$ wegen 3. Isosatz. $PN/N \cong P/(P\cup N) \Rightarrow PN/N$ ist Gruppe. \\ +$p \nmid [G:P] = \abs{G}_{p'} \Rightarrow [G : PN]$ ist Teiler von $m$, wird nicht von $p$ geteilt. \\ +$\Rightarrow PN/N \in \Syl_p(G/N)$. \\ +Nach 1.3.12 ist $[PN : P] = \frac{\abs{P} \cdot \abs{N}}{\abs{P \cap N}\abs{P}} = \frac{\abs{N}}{\abs{P \cap N}} \Rightarrow$ (nach oben) $P \cap N$ ist $p$-Untergruppe von $N$ mit $[N : P \cap N]$ wird nicht von $p$ geteilt. Also ist $P \cap N \in \Syl_p(N)$. \qed \\ +Vorsicht: $H \leq G \nRightarrow H \cap P \in \Syl_p(H)$ für $P \in \Syl_p(G)$ + +3.1.6 Satz: Sei $H \leq G, P \in \Syl_p(G)$. Dann gibt es $g \in G$ so, dass $g P g^{-1} \cap H \in \Syl_p(H)$ ist. \\ +Beweis: Sei $Q \in \Syl_p(H) \Rightarrow \exists P' \in \Syl_p(G)$ mit $Q \subseteq P' \Rightarrow \exists g \in G$ mit $P' = gPg^{-1} \cap H \supseteq Q$ \\ +Klar: $Q = P' \cap H$ (warum?) + \end{document} diff --git a/standard.tex b/standard.tex index f312aac..8f89514 100644 --- a/standard.tex +++ b/standard.tex @@ -87,6 +87,7 @@ \DeclareMathOperator{\PSL}{PSL} \DeclareMathOperator{\GF}{GF} \DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Syl}{Syl} %%