From 469790f78939f0f72ff6d725c30ae0faebb98c50 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 18 Jul 2023 23:59:10 +0200 Subject: [PATCH] feat(algebra1): aggiunge la nozione di normalizzatore ed il teorema di Cayley --- .../main.pdf | Bin 0 -> 193431 bytes .../main.tex | 47 ++++++++++++++++++ 2 files changed, 47 insertions(+) create mode 100644 Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.pdf create mode 100644 Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.tex diff --git a/Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.pdf b/Secondo anno/Algebra 1/5. 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Normalizzatore e teorema di Cayley/main.tex b/Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.tex new file mode 100644 index 0000000..5417e5a --- /dev/null +++ b/Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.tex @@ -0,0 +1,47 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Normalizzatore e teorema di Cayley} + \maketitle + + \begin{note} + Nel corso del documento per $(G, \cdot)$ si intenderà un qualsiasi gruppo. + \end{note} + + Sia $X = \{ H \subseteq G \mid H \leq G \}$ l'insieme dei sottogruppi di $G$. + Allora si può costruire un'azione $\varphi : G \to S(X)$ in modo tale che: + \[ g \xmapsto{\varphi} \left[ H \mapsto gHg\inv \right]. \] + Si definisce \textbf{normalizzatore} lo stabilizzatore di un sottogruppo + $H$ (e si indica con $N_G(H)$), mentre $\Orb(H)$ è l'insieme dei \textbf{coniugati} + di $H$. Si osserva in modo cruciale che $H \nsgeq G$ se e solo se + $\Orb(H) = \{H\}$, e quindi se e solo se $N_G(H) = G$. Analogamente si + osserva che $H$ è normale se e solo se: + \[ H = \bigcup_{h \in H} \Cl(h). \] \bigskip + + + Si illustra adesso un risultato principale della teoria dei gruppi che mette in + relazione ogni gruppo con il proprio gruppo di bigezioni, ed ogni gruppo finito con i + sottogruppi dei gruppi simmetrici. + + \begin{theorem}[di Cayley] + Ogni gruppo è isomorfo a un sottogruppo del suo gruppo di bigezioni. + In particolare, ogni gruppo finito $G$ è isomorfo a un sottogruppo di un gruppo + simmetrico. + \end{theorem} + + \begin{proof} + Si consideri l'azione\footnote{Tale azione prende il nome di \textbf{rappresentazione regolare a sinistra}. + Si può infatti definire un'azione analoga a destra ponendo $g \mapsto \left[ h \mapsto hg\inv \right]$, + costruendo dunque una \textit{rappresentazione regolare a destra}.} $\varphi : G \to S(G)$ tale per cui: + \[ g \xmapsto{\varphi} \left[ h \mapsto gh \right]. \] + Si mostra che $\varphi$ è fedele\footnote{L'azione $\varphi$ è molto + più che fedele; è infatti innanzitutto libera.}. Sia infatti $\varphi(g) = \Id$; allora + vale che $ge = e \implies g = e$. Quindi $\Ker \varphi$ è banale, e per il + Primo teorema di isomorfismo vale che: + \[ G \cong \Im \varphi \leq S(G). \] + Se $G$ è finito, $S(G)$ è isomorfo a $S_n$, dove $n := \abs{G}$, e quindi + $\Im \varphi$ è a sua volta isomorfo a un sottogruppo di $S_n$, da cui + la tesi. + \end{proof} +\end{document} \ No newline at end of file