From 8ba3f5fb75373184e7a074a1aaa32c094a5c7c59 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 18 Jul 2023 21:03:25 +0200 Subject: [PATCH] fix(algebra1): ripara gli enunciati del teorema di Cauchy --- .../4. Il teorema di Cauchy/main.pdf | Bin 193056 -> 191151 bytes .../4. Il teorema di Cauchy/main.tex | 42 +++++++++--------- 2 files changed, 20 insertions(+), 22 deletions(-) diff --git a/Secondo anno/Algebra 1/4. Il teorema di Cauchy/main.pdf b/Secondo anno/Algebra 1/4. 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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, + 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 elemento 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 + $h^{\nicefrac{o(h)}{p}}$ è un elemento 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 + induttiva, allora, esiste un elemento $tH$ di ordine $p$ in $G \quot H$. \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. + Pertanto, come prima, $t^{\nicefrac{o(t)}{p}}$ è un elemento 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 $\abs{H} = p$. + Sia $G$ un gruppo finito. Se un numero primo $p$ divide $\abs{G}$, allora + esiste $g \in G$ tale per cui $o(g) = p$. \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 $n = 1$, $G$ è ciclico e dunque ammette un generatore di ordine $p$, + completando il passo base. Sia ora $n > 1$ e si assuma che ogni gruppo di ordine $pk$ con + $k < n$ ammetta un elemento 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. + Se esiste un sottogruppo proprio $H < G$ tale per cui $p$ divide $\abs H$, allora $H$, e quindi + anche $G$, ammette un elemento 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: @@ -76,12 +75,12 @@ \[ \abs{Z(G)} \equiv 0 \pod p. \] 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 + $G$ è abeliano. Pertanto $G$ ammette un elemento 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 + Si mostra infine una dimostrazione alternativa del teorema di Cauchy (più immediata e facile da ricordare), basata su una particolare costruzione. \begin{proof}[Dimostrazione alternativa] @@ -89,8 +88,7 @@ \[ S = \{ (a_1, \ldots, a_p) \in G^p \mid a_1 \cdots a_p = e \}. \] 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 + $o(h) = p$ (infatti $h \neq e$), dimostrando la tesi. \medskip Si ipotizzi che tale elemento $h$ non esisti. Si consideri l'azione $\varphi$