From c9c6c616091846dcbad5c3af4c7dcc96fffa6361 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 18 Jul 2023 15:48:17 +0200 Subject: [PATCH] feat(algebra1): aggiunge appunti sull'azione di coniugio e i p-gruppi --- .../3. Azione di coniugio e p-gruppi/main.pdf | Bin 0 -> 157530 bytes .../3. Azione di coniugio e p-gruppi/main.tex | 95 ++++++++++++++++++ tex/latex/style/notes_2023.sty | 2 + 3 files changed, 97 insertions(+) create mode 100644 Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.pdf create mode 100644 Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex diff --git a/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.pdf b/Secondo anno/Algebra 1/3. 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Azione di coniugio e p-gruppi/main.tex b/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex new file mode 100644 index 0000000..be612c2 --- /dev/null +++ b/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex @@ -0,0 +1,95 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Azione di coniugio e $p$-gruppi} + \maketitle + + \begin{note} + Nel corso del documento per $(G, \cdot)$ si intenderà un qualsiasi gruppo. + \end{note} + + Si consideri l'omomorfismo $\zeta$ che associa ad ogni $g \in G$ l'automorfismo interno + che induce. Questo omomorfismo induce la cosiddetta: + + \begin{definition}[azione di coniugio] + Si definisce \textbf{azione di coniugio} l'azione di $G$ su sé stesso indotta da $\zeta : G \to \Aut(G)$ dove: + \[ g \xmapsto{\zeta} \varphi_g = \left[ h \mapsto g h g\inv \right]. \] + \end{definition} + + L'orbita di un elemento $g \in G$ prende in questo particolare caso il nome + di \textbf{classe di coniugio} (e si indica come $\Cl(g)$), mentre il suo stabilizzatore viene detto \textbf{centralizzatore} (indicato con $Z_G(g)$). Si verifica facilmente + che $Z_G(g)$ è composto da tutti gli elementi $h \in G$ che commutano con $g$, ossia + tali che $gh = hg$. Allora vale in particolare che: + \[ Z(G) = \Ker \zeta = \bigcap_{g \in G} Z_G(g). \] \medskip + + + Si osserva inoltre che se $g \in Z(G)$, allora $\Cl(g) = \{g\}$ (infatti, per $h \in G$, si avrebbe $h g h\inv = h h\inv g = g$). Si può dunque riscrivere la somma data dal + Teorema orbita-stabilizzatore nel seguente modo: + \[ \abs{G} = \sum_{g \in \mathcal{R}} \frac{\abs{G}}{\abs{Z_G(g)}} = \sum_{g \in Z(G)} \underbrace{\abs{\Cl(g)}}_{=1} + \sum_{g \in \mathcal{R} \setminus Z(G)} \frac{\abs{G}}{\abs{Z_G(g)}} = (*), \] + che riscritta ancora si risolve nella \textbf{formula delle classi di coniugio}: + \[ (*) = \abs{Z(G)} + \sum_{g \in \mathcal{R} \setminus Z(G)} \frac{\abs{G}}{\abs{Z_G(g)}}, \] + dove $\mathcal{R}$ è un insieme di rappresentanti delle orbite dell'azione di coniugio + (si osserva che ogni elemento di $Z(G)$ è un rappresentante dacché l'orbita di un + elemento del centro è banale). \medskip + + + Utilizzando la nozione di centralizzatore, si può contare ``facilmente'' il numero + di classi di coniugio di un gruppo. Infatti, si osserva crucialmente che + $\Fix(g)$ (il numero di elementi di $G$ lasciati invariati sotto il coniugio di $g$) + è lo stesso insieme $Z_G(g)$. Infatti vale che: + \[ \Fix(g) = \{ h \in G \mid gh = hg \} = Z_G(g). \] + Allora, per il lemma di Burnside, se $k(G)$ è il numero di classi di coniugio di $G$, vale che: + \[ k(G) = \frac{1}{\abs{G}} \sum_{g \in G} \abs{Z_G(g)}. \] \bigskip + + + La formula delle classi di coniugio risulta in particolare utile nella discussione + dei $p$-gruppi, definiti di seguito. + + \begin{definition}[$p$-gruppo] + Sia $G$ un gruppo finito. $G$ si dice allora \textbf{$p$-gruppo} se + $\abs{G} = p^n$ per $n \in \NN^+$ e un numero primo $p \in \NN$. + \end{definition} + + Infatti, grazie alla formula delle classi di coniugio, si osserva facilmente che il centro di un $p$-gruppo non è mai banale (ossia composto dalla sola identità), come mostra la: + + \begin{proposition} + Sia $G$ un $p$-gruppo. Allora $\abs{Z(G)} > 1$. + \end{proposition} + + \begin{proof} + Dalla formula delle classi di coniugio si ha che: + \[ \abs{G} = \abs{Z(G)} + \sum_{g \in \mathcal{R} \setminus Z(G)} \frac{\abs G}{\abs{Z_G(g)}}. \] + Si osserva in particolare che il secondo termine della somma a destra è divisibile + per $p$. Infatti, poiché $g \notin Z(G)$ per ipotesi, $Z_G(g) \neq Z(G)$; da cui + si deduce che $\abs{Z_G(g)}$ deve essere un divisore stretto di $p^n$, e dunque + che $p \mid \nicefrac{\abs G}{\abs{Z_G(g)}}$. Prendendo l'identità di sopra modulo + $p$, si deduce allora che: + \[ \abs{Z(G)} \equiv 0 \pod p. \] + Combinando questo risultato col fatto che $\abs{Z(G)} \geq 1$ (infatti $Z(G) \leq G$), + si conclude che deve valere necessariamente che $\abs{Z(G)} > 1$. + \end{proof} \medskip + + + Quest'ultima proposizione spiana il terreno per un risultato interessante sui + gruppi di ordine $p^2$, come mostra il: + + \begin{theorem} + Ogni gruppo $G$ di ordine $p^2$ è abeliano. + \end{theorem} + + \begin{proof} + Dal momento che $G$ è un $p$-gruppo, per la precedente proposizione + $\abs{Z(G)} > 1$. Allora $\abs{Z(G)}$ è pari a $p$ o $p^2$, per il + Teorema di Lagrange. Se $\abs{Z(G)}$ fosse pari a $p$, allora + $\abs{G \quot Z(G)} = \nicefrac{\abs G}{\abs{Z(G)}} = p$. Pertanto + $G \quot Z(G)$ sarebbe ciclico, e dunque $G$ sarebbe abeliano; assurdo, + dal momento che si era presupposto che $Z(G)$ fosse un sottogruppo proprio + di $G$, \Lightning. Allora $Z(G)$ ha ordine $p^2$, + e dunque $Z(G) = G$. + \end{proof} + + Questo risultato ha un'immediata applicazione, se combinato con il Teorema + fondamentale dei gruppi abeliani finiti. Infatti, esso implica che + $G$ deve per forza essere isomorfo a $\ZZ_{p^2}$ o a $\ZZ_{p} \times \ZZ_{p}$. +\end{document} \ No newline at end of file diff --git a/tex/latex/style/notes_2023.sty b/tex/latex/style/notes_2023.sty index 5233e57..137282e 100644 --- a/tex/latex/style/notes_2023.sty +++ b/tex/latex/style/notes_2023.sty @@ -209,6 +209,8 @@ % Spesso utilizzati durante il corso di Algebra 1 +\DeclareMathOperator{\Cl}{Cl} + \newcommand{\actson}{\circlearrowleft} % Comandi personali.