From 5c7c8085a6ac4478fb7a67fd545d3158401a8683 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 6 Oct 2023 20:05:32 +0200 Subject: [PATCH] feat(algebra1): aggiunge i tre teoremi di isomorfismo --- .../9. I teoremi di isomorfismo/main.pdf | Bin 0 -> 177187 bytes .../9. I teoremi di isomorfismo/main.tex | 135 ++++++++++++++++++ 2 files changed, 135 insertions(+) create mode 100644 Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.pdf create mode 100644 Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.tex diff --git a/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.pdf b/Secondo anno/Algebra 1/9. 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I teoremi di isomorfismo/main.tex b/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.tex new file mode 100644 index 0000000..8c839f9 --- /dev/null +++ b/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.tex @@ -0,0 +1,135 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{I teoremi di isomorfismo} + \maketitle + + \begin{note} + Nel corso del documento per $(G, \cdot)$ si intenderà un qualsiasi gruppo. Analogamente si intenderà lo stesso per + $G'$. + \end{note} + + Si illustrano i tre teoremi di isomorfismo nella loro + forma più generale. + + \begin{theorem}[Primo teorema di isomorfismo] + Sia $\varphi$ un omomorfismo da $G$ in $G'$. Allora, + se $N \leq \Ker \varphi$, esiste un unico omomorfismo + $f$ da $G \quot N$ in $G'$ che faccia commutare il + seguente diagramma commutativo: + \[\begin{tikzcd} + G &&& {G'} \\ + \\ + {G/N} + \arrow["{\pi_N}"', two heads, from=1-1, to=3-1] + \arrow["\varphi", from=1-1, to=1-4] + \arrow["f"', from=3-1, to=1-4] + \end{tikzcd}\] + Inoltre, tale $f$ è iniettiva se e solo se $N = \Ker \varphi$ + e in tal caso induce il seguente isomorfismo: + \[ G \quot {\Ker \varphi} \cong \Im \varphi. \] + \end{theorem} + + \begin{proof} + Affinché il diagramma commuti, deve valere la seguente + relazione: + \[ \varphi(g) = f(\pi_N(g)) = f(gN). \] + Pertanto l'unica possibilità è che valga $f(gN) = \varphi(g)$. + Chiaramente tale mappa è ben definita, infatti se $n \in N$, + $\varphi(gn) = \varphi(g) \varphi(n) = \varphi(g)$, dacché + $n$ in particolare è anche un elemento di $\Ker \varphi$. + Inoltre $f$ è un omomorfismo, dal momento che + $f(gN hN) = f(ghN) = \varphi(gh) = \varphi(g) \varphi(h) = + f(gN) f(hN)$. \medskip + + + Sia $k \in \Ker \varphi$. Se $f$ è iniettiva, allora $f(gN) = \varphi(g) = e \implies gN = N$. Dal momento che + $f(kN) = \varphi(k) = e$, $kN = N$, e quindi $k \in N$, + da cui si deduce che $N = \Ker \varphi$. Se invece + $N = \Ker \varphi$, $f(gN) = e \implies \varphi(g) = e \implies g \in N$, e quindi $gN = N$, l'identità di + $G \quot N$, da cui si deduce che $f$ è iniettiva. In tal + caso la restrizione sull'immagine di $f$ a + $\Im f$, coincidente con $\Im f \circ \pi_N = \Im \varphi$ + dacché $\pi_N$ è surgettiva, fornisce l'isomorfismo + ricercato. + \end{proof} + + In particolare si osserva che $\Ker f = \Ker \varphi \quot N$, + infatti: + \[ \Ker f = \{ gN \mid \varphi(g) = e \} = \{ gN \mid g \in \Ker \varphi \} = \Ker \varphi \quot N. + \] + + \begin{theorem}[Secondo teorema di isomorfismo, o teorema del diamante] + Siano $H$, $N \leq G$ con $N \nsgeq G$. Allora\footnote{ + Si osserva che effettivamente $H \cap N$ è normale in + $H$. Infatti se $g \in H \cap N$, allora, se + $h \in H$, $h g h\inv$ appartiene sempre a $N$ + perché $N$ è normale in $G$ e appartiene anche + ad $H$ poiché è prodotto di elementi in $H$. + }\footnote{ + Analogamente $N$ è normale in $HN$, essendo + normale in $G$. + }: + \[ H \quot (H \cap N) \cong HN \quot N. \] + Pertanto se si considera il seguente diagramma: + \[\begin{tikzcd}[column sep=0em,row sep=3em] + & HN \arrow[dl,dash] \arrow[dr,dash] \\ + H \arrow[dr,dash] && N \arrow[dl,dash] \\ + & H\cap N + \end{tikzcd}\] + i lati paralleli del parallelogramma (``diamante'') + forniscono gli isomorfismi dell'enunciato se anche + $H$ è normale in $G$. + \end{theorem} + + \begin{proof} + Si costruisce l'omomorfismo $\varphi : H \to HN \quot N$ + tale per cui $h \mapsto hN$. Si osserva che $\varphi$ è + effettivamente un omomorfismo, infatti: + \[ \varphi(hh') = (hh')N = (hN) (h'N) = \varphi(h) \varphi(h'). \] + + Sia $hnN \in HN \quot N$. Allora $hnN = hN$, e quindi + $\varphi(h) = hN = hnN$, da cui si deduce che + $\varphi$ è surgettiva (e quindi $\Im \varphi = HN \quot N$). + \medskip + + + Sia $\varphi(h) = e$. Allora $hN = N \implies h \in H \cap N$. + Si deduce dunque che $\Ker \varphi = H \cap N$, da cui, + applicando il Primo teorema di isomorfismo, si ottiene + la tesi: + \[ H \quot (H \cap N) \cong HN \quot N. \] + \end{proof} + + \begin{theorem}[Terzo teorema di isomorfismo] + Siano $H$ e $N$ due sottogruppi normali di $G$ e sia + $N \leq H$. Allora\footnote{ + Ci sono più modi per vedere che $H \quot N$ è + normale in $G \quot N$. Un modo di vederlo si + ottiene dalla dimostrazione stessa del teorema, + dal momento che si ottiene che $H \quot N$ è + il kernel dell'omomorfismo $\varphi$. Altrimenti, + se $hN \in H \quot N$, $gN hN g\inv N = (ghg\inv)N$, + e poiché $H$ è normale in $G$, $ghg\inv \in H$, da + cui $(ghg\inv)N \in H \quot N$. + }: + \[ \frac{G \quot N}{H \quot N} \cong G \quot H. \] + \end{theorem} + + \begin{proof} + Si costruisce l'omomorfismo $\varphi : G \quot N \to G \quot H$ tale per cui $gN \mapsto gH$. Si verifica innanzitutto + che la mappa $\varphi$ è ben definita: + \[ gnH = gH \impliedby N \subseteq H. \] + Inoltre $\varphi$ è effettivamente un omomorfismo dal momento + che: + \[ \varphi(gkN) = gkH = gH \, kH = \varphi(gN) \varphi(kN). \] + Chiaramente $\varphi$ è una mappa surgettiva e quindi + $\Im \varphi = G \quot H$. + Allora, se $g \in \Ker \varphi$, $\varphi(gN) = gH = H$, e quindi $g \in H$. Pertanto $\Ker \varphi = \{ + gN \mid g \in H + \} = H \quot N$. Si conclude allora, per il Primo teorema + di isomorfismo, che: + \[ \frac{G \quot N}{H \quot N} \cong G \quot H. \] + \end{proof} +\end{document} \ No newline at end of file