From c65a0c5429756914aeff7e0b0469afec49b5fde3 Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 20 Jul 2023 18:09:41 +0200 Subject: [PATCH] feat(algebra1): aggiunge la dimostrazione del teorema di corrispondenza --- .../main.pdf | Bin 0 -> 156852 bytes .../main.tex | 96 ++++++++++++++++++ 2 files changed, 96 insertions(+) create mode 100644 Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.pdf create mode 100644 Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.tex diff --git a/Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.pdf b/Secondo anno/Algebra 1/7. 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Il teorema di corrispondenza e catene di sottogruppi normali/main.tex b/Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.tex new file mode 100644 index 0000000..14e20ec --- /dev/null +++ b/Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.tex @@ -0,0 +1,96 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Il teorema di corrispondenza e catene di sottogruppi normali} + \maketitle + + Si illustra adesso un teorema che mette in corrispondenza + i sottogruppi di $G \quot H$ con i sottogruppi di $G$ + che contengono $H$. Benché questo teorema possa sembrare + a prima vista di poca utilità, in realtà svela alcune + proprietà che hanno portato allo sviluppo della celebre + teoria di Galois. Non solo, guardando anche nelle piccole + applicazioni, il teorema di corrispondenza permette di + contare molto facilmente i sottogruppi di $G \quot H$, + nonché di dimostrare l'esistenza di una catena di + $p$-sottogruppi normali contenente tutti gli ordini + possibili per un $p$-gruppo. + + \begin{theorem}[di corrispondenza] + Sia $H$ un sottogruppo normale di $G$. Allora + la proiezione al quoziente $\pi_H : G \to G \quot H$ + induce una bigezione tra l'insieme + \[ X = \{ K \leq G \mid H \subseteq K \} \] + dei sottogruppi di $G$ che contengono $H$ e l'insieme + \[ Y = \{ K' \leq G \quot H \} \] + dei sottogruppi di $G \quot H$. Tale bigezione preserva + la normalità di un gruppo e il suo indice, ossia: + \begin{itemize} + \item $K \nsgeq G \iff K' \nsgeq G \quot H$, + \item $\left[ G : K \right] = \left[ G \quot H : K' \right]$, + \end{itemize} + dove $K \in X$ e $K' \in Y$ sono in corrispondenza biunivoca + mediante $\pi_H$. + \end{theorem} + + \begin{proof} + Sia $\alpha : X \to Y$ definita nel seguente modo: + \[ K \xmapsto{\alpha} \pi_H(K), \] + dove si osserva che $\pi_H(K) = \{ kH \mid k \in K \} = K \quot H \leq G \quot H$. + Si definisce analogamente $\beta : Y \to X$ in modo tale che: + \[ K' \xmapsto{\beta} \pi_H\inv(K'). \] + Le due mappe sono entrambe ben definite (infatti $\pi_H\inv(K')$ è sempre un sottogruppo di $G$ e contiene + sempre $H$, dacché $H \in K'$, essendo l'identità di $G \quot H$). + È dunque sufficiente mostrare che vale $\beta \circ \alpha = \Id_X$ e che $\alpha \circ \beta = \Id_Y$. \bigskip + + + Siano quindi $K \in X$ e $K' \in Y$. Chiaramente + $\pi_H(\pi_H\inv(K')) = K'$, dal momento che $\pi_H$ è + surgettiva; dunque $\alpha \circ \beta = \Id_Y$. Inoltre + $\pi_H \inv (\pi_H(K)) = \pi_H\inv (K \quot H) = \{ g \in G \mid gH \in K \quot H \} = K$\footnote{ + Infatti se $gH=kH$ con $k \in K$, esiste un $h \in H$ tale per cui $g=kh$. + Dal momento che $H \subseteq K$, $g$ è dunque un elemento di + $K$.}, da cui $\beta \circ \alpha = \Id_X$. Quindi $X$ e $Y$ sono in corrispondenza biunivoca + tramite $\alpha$ e $\beta$. \bigskip + + + Rimane da dimostrare che $\alpha$ e $\beta$ preservano + la normalità e l'indice di sottogruppo. Se $K \nsgeq G$, + allora chiaramente $K' = K \quot H \nsgeq G \quot H$ + (infatti $gH \, kH \, g\inv H = (gkg\inv) H$, dove + $gkg\inv \in K$ per ipotesi di normalità). Sia + ora $K' \nsgeq G \quot H$. Allora, se $k \in K$, + $gH \, kH \, g\inv H = (gkg\inv) H$, e per ipotesi + di normalità deve esistere $k' \in K$ tale per cui + $(gkg\inv) H = k'H$, e quindi deve esistere + $h \in H$ tale per cui $gkg\inv = k'h$. Dal momento + che $H \subseteq K$, $gkg\inv \in K$, e quindi + $K \nsgeq G$. \bigskip + + + Per mostrare che l'indice di sottogruppo si preserva + si dimostra che esiste lo stesso numero di classi + laterali in $G \quot K$ e $(G \quot H) \quot (K \quot H)$. + Pertanto è sufficiente mostrare che: + \[ + xK = yK \iff xH (K \quot H) = yH (K \quot H), \qquad x, y \in G. + \] + Infatti, in tal caso vi sarebbero esattamente $[G : K]$ + classi laterali in $(G \quot H) \quot (K \quot H)$. + Si consideri ora la classe laterale $xH(K \quot H)$: + \[ + xH(K \quot H) = \{ xHkH \mid k \in K \} = \{ (xk) H \mid k \in K \}, + \] + dove nell'ultima uguaglianza si è impiegata la normalità + di $H$ in $G$ (altrimenti il prodotto non sarebbe ben + definito). + Analogamente $yH(K \quot H) = \{ (yk)H \mid k \in K \}$. + Quindi, se $xH (K \quot H) = yH (K \quot H)$, allora $xH = (yk)H$, con $k \in K$. + Allora $x = ykh$ con $h \in H$. Poiché $H \subseteq K$, si deduce + quindi che $xK = yK$. Infine, se $xK = yK$, esiste + $k \in K$ tale per cui $x=yk$. Allora: + \[ xH(K \quot H) = yH \, kH (K \quot H) = yH(K \quot H), \] + da cui la tesi. + \end{proof} +\end{document} \ No newline at end of file