From 487570d4e9dc6940240d3565f48e2f5b5f23b38d Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 18 Jul 2023 14:49:42 +0200 Subject: [PATCH] feat(algebra1): aggiunge appunti sulle azioni di gruppo --- .../main.pdf | Bin 0 -> 184025 bytes .../main.tex | 149 ++++++++++++++++++ tex/latex/style/notes_2023.sty | 4 + 3 files changed, 153 insertions(+) create mode 100644 Secondo anno/Algebra 1/2. Azione di un gruppo su un insieme/main.pdf create mode 100644 Secondo anno/Algebra 1/2. Azione di un gruppo su un insieme/main.tex diff --git a/Secondo anno/Algebra 1/2. Azione di un gruppo su un insieme/main.pdf b/Secondo anno/Algebra 1/2. 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Azione di un gruppo su un insieme/main.tex b/Secondo anno/Algebra 1/2. Azione di un gruppo su un insieme/main.tex new file mode 100644 index 0000000..00edadd --- /dev/null +++ b/Secondo anno/Algebra 1/2. Azione di un gruppo su un insieme/main.tex @@ -0,0 +1,149 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Azione di un gruppo su un insieme} + \maketitle + + \begin{note} + Nel corso del documento per $(G, \cdot)$ si intenderà un qualsiasi gruppo. + Si scriverà $gh$ per indicare $g \cdot h$, omettendo il punto. Analogamente + con $X$ si indicherà un insieme generico qualsiasi. + \end{note} + + \begin{definition}[azione di un gruppo su un insieme] Sia $X$ un insieme. Allora + un'applicazione $\varphi : G \to S(X)$ tale che $g \xmapsto{\varphi} \left[ x \mapsto g \cdot x \right]$ si dice \textbf{azione di $G$ su $X$} + se è un omomorfismo di gruppi. + \end{definition} \medskip + + + Se $G$ agisce tramite $\varphi$ su $X$, si dice allora che $X$ è un $G$-insieme. Si dice inoltre che l'azione $\varphi$ è \textbf{fedele} se $\varphi$ è iniettiva, + ossia se e solo se $\varphi(g) = \Id \implies g = e$. + + \begin{definition}[stabilizzatore] Sia $x \in X$. Allora si definisce lo + \textbf{stabilizzatore di $x$}, denotato come $\Stab(x)$, come il sottogruppo di $G$ + tale per cui: + \[ \Stab(x) = \{ g \in G \mid g \cdot x = x \}. \] + \end{definition} \medskip + + + Si può allora constatare che $\varphi$ è fedele se e solo se: + \[ \Ker \varphi = \bigcap_{x \in X} \Stab(x) = \{e\}. \] \medskip + + + Si costruisce adesso una relazione di equivalenza $\sim$ su $X$, data dalla + seguente definizione: + \[ x \sim y \defiff \exists g \in G \mid g \cdot x = y. \] + Le classi di equivalenza di $\sim$ vengono dette $\textbf{orbite}$ e si + pone $\Orb(x) := \left[ x \right]_\sim$. \medskip + + \begin{definition}[azione libera] + Si dice che $\varphi$ è un'azione libera (o che $G$ agisce liberamente + su $X$) se $\Stab(x) = \{e\}$ per ogni scelta di $x \in X$. + \end{definition} + + \begin{definition}[azione transitiva] + Si dice che $\varphi$ è un'azione transitiva (o che $G$ agisce transitivamente + su $X$) se esiste un'unica classe di equivalenza di $\sim$ (ossia se + $\forall x$, $y \in X$, $\exists g \in G \mid g \cdot x = y$). In tal + caso si dice che $X$ è un $G$-insieme omogeneo. + \end{definition} + + \begin{definition}[azione semplicemente transitiva] + Si dice che $\varphi$ è un'azione semplicemente transitiva (o che $G$ agisce in + maniera semplicemente transitiva su $X$) se $\varphi$ è un'azione libera e + transitiva. In tal caso si dice che $X$ è un $G$-insieme omogeneo principale. + \end{definition} \medskip + + In generale, un'azione può essere solamente libera o solamente transitiva. Chiaramente + però la libertà di un'azione ne implica la fedeltà, e non il contrario. Tuttavia + nel caso particolare dei gruppi abeliani, la fedeltà e la transitività di un'azione + ne implicano anche la libertà, come enunciato dalla: + + \begin{proposition} + Sia $G$ abeliano. Allora, se $\varphi$ è fedele e transitiva, $\varphi$ è + semplicemente transitiva. + \end{proposition} + + \begin{proof} + È sufficiente dimostrare che $\varphi$ è anche libera, ossia che $\Stab(x) = \{e\}$ + per ogni scelta di $x \in X$. Sia allora $g \in \Stab(x)$. Si mostra che + $g \in \Ker \varphi$, da cui si dedurrà che $g = e$. \medskip + + + Sia $y \in X$. Poiché $\varphi$ è transitiva, $x \sim y$, e quindi esiste + $h \in G$ tale per cui $h \cdot x = y$. Pertanto, sfruttando la commutatività + di $G$, $g \cdot y = g \cdot (h \cdot x) = + h \cdot (g \cdot x) = h \cdot x = y$, da cui si deduce che $\varphi(g) = \Id$, + concludendo la dimostrazione. + \end{proof} \bigskip + + Si dimostra adesso il teorema più importante sulle azioni di gruppi sugli insiemi: il + Teorema orbita-stabilizzatore, un ``analogo'' del Primo teorema di isomorfismo per + le azioni\footnote{Si lascia al lettore la gioia di dimostrare il Primo teorema di isomorfismo proprio a partire dal Teorema orbita-stabilizzatore (indizio: se $f \in \Hom(G,H)$, si può considerare l'azione $\varphi : G \to S(H)$ tale che $g \xmapsto{\varphi} \left[ h \mapsto g \cdot h = f(g)h \right]$). Si noterà infatti + che la dimostrazione del Teorema orbita-stabilizzatore ricalca totalmente la + stessa idea della dimostrazione del Primo teorema di isomorfismo.}. + + \begin{theorem}[orbita-stabilizzatore] + Sia $x \in X$. Allora la mappa $\alpha : G \quot \Stab(x) \to \Orb(x)$ tale + che $g \Stab(x) \xmapsto{\alpha} g \cdot x$ è una bigezione. + \end{theorem} + + \begin{proof} + Si mostra che la mappa $\alpha$ è ben definita. Se $g \in G$ e $s \in \Stab(x)$, + allora $\alpha(gs \Stab(x)) = (gs) \cdot x = g \cdot x = \alpha(g \Stab(x))$. \medskip + + + Si dimostra allora l'iniettività di $\alpha$. Siano $g$ e $h \in G$ tali + che $\alpha(g \Stab(x)) = \alpha(h \Stab(x))$. Allora $g \cdot x = h \cdot x \implies + (h\inv g) \cdot x = x \implies h \inv g \in \Stab(x)$; pertanto $h \in g \Stab(x) \implies g \Stab(x) = h \Stab(x)$, da cui l'iniettività. \medskip + + + Infine si mostra la surgettività di $\alpha$. Se $y \in \Orb(x)$, allora esiste + $g \in G$ tale per cui $g \cdot x = y$, e quindi $\alpha(g \Stab(x)) = g \cdot x = y$, + da cui la surgettività. + \end{proof} \bigskip + + + Se $G$ è finito, il Teorema orbita-stabilizzatore implica anche un'identità aritmetica + riguardante le cardinalità di $\Stab(x)$ e $\Orb(x)$: + \[ \abs G = \abs{\Stab(x)} \abs{\Orb(x)}. \] + Da questa identità si può estrarre un'ulteriore uguaglianza: + \[ \abs X = \sum_{x \in \mathcal{R}} \abs{\Orb(x)} = \sum_{x \in \mathcal{R}} \frac{\abs G}{\abs{\Stab(x)}}, \] + dove $\mathcal{R}$ è un insieme dei rappresentanti delle orbite dell'azione. Questo + fatto è un'immediata conseguenza del fatto che la relazione $\sim$ è di equivalenza, + e che, in quanto tale, induce una partizione dell'insieme $X$ mediante i suoi + rappresentanti: + \[ X = \bigsqcup_{x \in \mathcal{R}} \Orb(x). \] \bigskip + + + Si introduce adesso il concetto di \textit{punti fissi} di un dato $g \in G$, a cui + seguirà il \textit{lemma di Burnside}, un risultato utile per contare il numero + di orbite di un'azione. + + \begin{definition}[punti fissi di $g$] + Si definisce l'insieme $\Fix(g)$ come il sottoinsieme di $X$ dei punti + lasciati fissi da $g$, ossia: + \[ \Fix(g) = \{ x \in X \mid g \cdot x = x \}. \] + \end{definition} + + \begin{proposition}[lemma di Burnside] + $\abs{X \quot \sim} = \frac{1}{\abs{G}} \sum_{g \in G} \abs{\Fix(g)}$. + \end{proposition} + + \begin{proof} + L'idea chiave risiede nell'osservare che $\sum_{g \in G} \abs{\Fix(g)}$ conta + gli elementi dell'insieme $S$, dove: + \[ S = \{ (g, x) \in G \times X \mid g \cdot x = x \} \subseteq G \times X. \] + Infatti, gli stessi elementi sono contati da $\sum_{x \in X} \abs{\Stab(x)}$. + Applicando allora il Teorema orbita-stabilizzatore, ed indicando + con $\mathcal{R}$ un insieme dei rappresentanti delle orbite, la + somma si riscrive come: + \[ \sum_{g \in G} \abs{\Fix(g)} = \sum_{x \in X} \abs{\Stab(x)} = + \sum_{r \in \mathcal{R}} \sum_{x \in \Orb(r)} \abs{\Stab(x)} = (*), + \] + a sua volta riscritta come: + \[ (*) = \sum_{r \in \mathcal{R}} \sum_{x \in \Orb(r)} \frac{\abs{G}}{\abs{\Orb(x)}} = \sum_{r \in \mathcal{R}} \sum_{x \in \Orb(r)} \frac{\abs{G}}{\abs{\Orb(r)}} = \abs{G} \abs{X \quot \sim}, \] + dove è stato cruciale osservare che, per $x \in \Orb(r)$, $\Orb(x) = \Orb(r)$. + \end{proof} +\end{document} diff --git a/tex/latex/style/notes_2023.sty b/tex/latex/style/notes_2023.sty index 86c99d1..5233e57 100644 --- a/tex/latex/style/notes_2023.sty +++ b/tex/latex/style/notes_2023.sty @@ -207,6 +207,10 @@ \newcommand{\mapstoby}[1]{\xmapsto{#1}} \newcommand{\oplusperp}{\oplus^\perp} +% Spesso utilizzati durante il corso di Algebra 1 + +\newcommand{\actson}{\circlearrowleft} + % Comandi personali. \newcommand{\card}[1]{\left|#1\right|}