From 8f79b19a68c8e76da9fcbb118cd7a78dbd42769f Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 18 Jul 2023 18:41:13 +0200 Subject: [PATCH] feat(algebra1): aggiunge gli appunti sul teorema di Cauchy --- .../4. Il teorema di Cauchy/main.pdf | Bin 0 -> 193056 bytes .../4. Il teorema di Cauchy/main.tex | 117 ++++++++++++++++++ tex/latex/style/notes_2023.sty | 1 + 3 files changed, 118 insertions(+) create mode 100644 Secondo anno/Algebra 1/4. Il teorema di Cauchy/main.pdf create mode 100644 Secondo anno/Algebra 1/4. Il teorema di Cauchy/main.tex diff --git a/Secondo anno/Algebra 1/4. Il teorema di Cauchy/main.pdf b/Secondo anno/Algebra 1/4. 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Il teorema di Cauchy/main.tex b/Secondo anno/Algebra 1/4. Il teorema di Cauchy/main.tex new file mode 100644 index 0000000..78ef66b --- /dev/null +++ b/Secondo anno/Algebra 1/4. Il teorema di Cauchy/main.tex @@ -0,0 +1,117 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Il teorema di Cauchy} + \maketitle + + \begin{note} + Nel corso del documento per $(G, \cdot)$ si intenderà un qualsiasi gruppo. + \end{note} + + Si dimostra in questo documento, per ben due volte, un inverso parziale + del teorema di Lagrange, il celebre teorema di Cauchy. Tale teorema + asserisce che se $p$ è un numero primo che divide l'ordine di $G$, + allora esiste un sottogruppo $H$ di $G$ di ordine $p$. \medskip + + + Si mostra innanzitutto che il teorema vale per gruppi abeliani. + + \begin{theorem}[di Cauchy per gruppi abeliani] + Sia $G$ un gruppo abeliano finito. Se $p$ divide $\abs{G}$, allora + esiste $H \leq G$ tale per cui $\abs{H} = p$. + \end{theorem} + + \begin{proof} + Sia $\abs{G} = pn$ con $n \in \NN^+$. Si dimostra per induzione su $n$ la + validità della tesi. Se $n = 1$, allora $G$ stesso è un sottogruppo di ordine $p$, completando il passo base. \medskip + + + Sia allora $n > 1$ e si ipotizzi allora che tutti i gruppi tali che $\abs{G} = pk$ con $k < n$, $k \in \NN^+$ ammettano un sottogruppo di ordine $p$. Sia $h \in G$, $h \neq e$ (questo $h$ sicuramente esiste, dal momento che $p > 1$). Se $p \mid o(h)$, allora + $\Cyc{h^{\nicefrac{o(h)}{p}}}$ è un sottogruppo di $G$ di ordine $p$. Altrimenti, + si consideri $H = \Cyc{h}$. \medskip + + + Dal momento che $G$ è abeliano, $H$ è normale, e dunque si può considerare il gruppo quoziente $G \quot H$. Poiché $p \nmid o(h) = \abs H$ e $p$ divide $\abs G$, + $p$ divide anche $\abs{G \quot H}$ per il teorema di Lagrange. Inoltre, poiché + $o(h) > 1$ (infatti $h \neq e$), $\abs{G \quot H} < \abs{G}$. Per l'ipotesi + induttiva, allora, esiste un sottogruppo $T$ di ordine $p$ di $G \quot H$. Poiché + $T$ è di ordine $p$, $T$ è ciclico, e quindi esiste $t \in G$, $t \neq e$ tale per cui + $T = \Cyc{tH}$. \medskip + + + Si mostra adesso che $p \mid o(t)$. Si consideri la proiezione al quoziente $\pi : G \to G \quot H$ tale per cui: + \[ g \xmapsto{\pi} gH. \] + Allora $p = o(tH) \mid o(t)$, dal momento che $eH=\pi(t^{o(t)})=(tH)^{o(t)}$. + Pertanto, come prima, si può estrarre da $\Cyc{t}$ un sottogruppo di $G$ + di ordine $p$, concludendo il passo induttivo. + \end{proof} \bigskip + + Di seguito si dimostra il teorema di Cauchy in generale. + + \begin{theorem}[di Cauchy] + Sia $G$ un gruppo finito. Se $p \mid \abs{G}$, allora + esiste $H \leq G$ tale per cui $\abs{H} = p$. + \end{theorem} + + \begin{proof} + Sia $\abs{G} = pn$ con $n \in \NN^+$. Si dimostra la tesi per induzione. + Se $n = 1$, $G$ stesso è sottogruppo di ordine $p$, completando il passo + base. Sia ora $n > 1$ e si assuma che ogni sottogruppo di ordine $pk$ con + $k < n$ ammetta un sottogruppo di ordine $p$. \medskip + + + Se esiste $H \lneq G$ tale per cui $p$ divide $\abs H$, allora $H$, e quindi + anche $G$, ammette un sottogruppo di ordine $p$ per l'ipotesi induttiva. + Si assuma dunque che non esiste alcun sottogruppo proprio $H < G$ tale + per cui $p$ divide $\abs H$. Si consideri la formula delle classi + di coniugio: + \[ \abs G = \abs{Z(G)} + \sum_{g \in \mathcal{R} \setminus Z(G)} \frac{\abs G}{\abs{Z_G(g)}}, \] + dove $\mathcal{R}$ è un insieme dei rappresentanti delle classi di coniugio + di $G$. Se $g \in \mathcal{R} \setminus Z(G)$, allora $Z_G(g)$ è un sottogruppo + proprio di $G$, e quindi, per ipotesi, $p$ non divide $\abs{Z_G(g)}$; e quindi + $p$ divide ancora $\nicefrac{\abs{G}}{\abs{Z_G(g)}}$ (e quindi il secondo termine + del secondo membro). Allora, prendendo + l'identità modulo $p$, si deduce che: + \[ \abs{Z(G)} \equiv 0 \pod p. \] + Poiché $Z(G)$ è un sottogruppo di $G$, se valesse $Z(G) < G$, si violerebbero + le ipotesi iniziali. Pertanto deve necessariamente valere $Z(G) = G$, e quindi + $G$ è abeliano. Pertanto $G$ ammette un sottogruppo di ordine $p$ per il Teorema + di Cauchy per i gruppi abeliani; completando il passo induttivo. + \end{proof} \smallskip + + + Si mostra inoltre una dimostrazione alternativa del teorema di Cauchy (più immediata + e facile da ricordare), basata su una particolare costruzione. + + \begin{proof}[Dimostrazione alternativa] + Si consideri l'insieme $S$, dove: + \[ S = \{ (a_1, \ldots, a_p) \in G^p \mid a_1 \cdots a_p = e \}. \] + Dimostrando che esiste un elemento $h \in G$ diverso dall'identità tale + per cui $(h, \ldots, h) \in S$, si mostra che $h^p = e$, e dunque che + $o(h) = p$ (infatti $h \neq e$). Allora, in tal caso, $\Cyc{h}$ è + un sottogruppo di $G$ di ordine $p$, e si dimostra la tesi. \medskip + + + Si ipotizzi che tale elemento $h$ non esisti. Si consideri l'azione $\varphi$ + di $\ZZ \quot p\ZZ$ su $S$ univocamente determinata\footnote{$\ZZ \quot p\ZZ$ è infatti generato da $1$.} dalla relazione: + \[ 1 \xmapsto{\varphi} \left[ (a_1, a_2, \ldots, a_p) \mapsto (a_2, \ldots, a_p, a_1) \right]. \] + In particolare $m \cdot (a_1, \ldots, a_p)$ restituisce una $p$-upla ottenuta + ``ciclando a sinistra'' la $p$-upla iniziale di $m$ posizioni. Si consideri la + somma data dal teorema orbita-stabilizzatore: + \[ \abs{S} = \sum_{x \in S} \frac{p}{\abs{\Stab(x)}} = 1 + \sum_{x \in S \setminus \{(e,\ldots,e)\}} \frac{p}{\abs{\Stab(x)}}. \] + Poiché $\Stab(x) \leq \ZZ \quot p\ZZ$, gli unici ordini di $\Stab(x)$ possono + essere $1$ e $p$. Se tuttavia, per $x \in S \setminus \{(e,\ldots,e)\}$, + valesse $\Stab(x) = \ZZ \quot p\ZZ$, $x$ avrebbe coordinate tutte uguali, + e quindi, per ipotesi, $x = (e,\ldots,e)$, \Lightning. Quindi il secondo + termine del secondo membro vale esattamente $pk$, dove $k = \abs{S \setminus \{(e,\ldots,e)\}}$. \medskip + + + Si osserva adesso che $\abs S = n^{p-1}$, dove $n = \abs G$. Infatti è sufficiente + determinare le prime $p-1$ coordinate, per le quali vi sono $n$ scelte, per determinare + anche l'ultima coordinata tramite la relazione $a_1 \cdots a_n = e$. Prendendo + allora la precedente identità modulo $p$, si ottiene: + \[ 1 \equiv 0 \pod p, \] + da cui l'assurdo ricercato, \Lightning. + \end{proof} +\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 137282e..ef20743 100644 --- a/tex/latex/style/notes_2023.sty +++ b/tex/latex/style/notes_2023.sty @@ -212,6 +212,7 @@ \DeclareMathOperator{\Cl}{Cl} \newcommand{\actson}{\circlearrowleft} +\newcommand{\Cyc}[1]{\left<#1\right>} % Comandi personali.