From 3e4ad0c63fbb109ca86ef4a8876ca6f79b9be810 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 20 Oct 2023 17:52:41 +0200 Subject: [PATCH] feat(algebra1): inizializza gli appunti sul prodotto semidiretto --- .../12. Il prodotto semidiretto/main.pdf | Bin 0 -> 150364 bytes .../12. Il prodotto semidiretto/main.tex | 54 ++++++++++++++++++ 2 files changed, 54 insertions(+) create mode 100644 Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.pdf create mode 100644 Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex diff --git a/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.pdf b/Secondo anno/Algebra 1/12. 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Il prodotto semidiretto/main.tex b/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex new file mode 100644 index 0000000..f7c2cd2 --- /dev/null +++ b/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex @@ -0,0 +1,54 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Il prodotto semidiretto} + \maketitle + + \begin{note} + Nel corso del documento con $G$ un qualsiasi gruppo. + \end{note} + + Siano $H$ e $K$ due gruppi. Allora, dato un omomorfismo $\varphi : K \to \Aut(H)$ e + detto $\varphi_k := \varphi(k)$, + si può costruire un gruppo su $H \times K$ detto \textbf{prodotto semidiretto} tra $H$ e $K$, + indicato con $H \rtimes_\varphi K$, dove l'operazione è data da: + \[ (h,k)(h',k') = (h \, \varphi_k(h'), k k'). \] + + In questo gruppo l'inverso di $(h, k)$ è dato da $(\varphi_k\inv(h\inv), k\inv)$, + infatti: + \[ (h, k) (\varphi_k\inv(h\inv), k\inv) = (h \, \varphi_k(\varphi_k\inv(h\inv)), kk\inv) = (e, e). \] + In particolare, se $\varphi$ è banale, e quindi $k \xmapsto{\varphi} \Id_H$, + $H \rtimes_\varphi K$ ha la stessa struttura usuale del prodotto diretto. Nel + prodotto semidiretto $H \rtimes_\varphi K$ si possono identificare facilmente + $H$ e $K$ nei sottogruppi $H \times \{e\}$ e $\{e\} \times K$. \medskip + + + Detto $\alpha: H \rtimes_\varphi K \to K$ la mappa che associa $(h, k)$ a + $k$, si verifica che $\alpha$ è un omomorfismo con $\Ker \alpha = H \times \{e\}$. + Pertanto $H \times \{e\}$ è un sottogruppo normale di $H \rtimes_\varphi K$, + mentre in generale $K \times \{e\}$ non lo è. \medskip + + + Si illustra adesso un teorema che permette di decomporre, sotto opportune ipotesi, + un gruppo in un prodotto semidiretto di due suoi sottogruppi: + + \begin{theorem}[di decomposizione in prodotto semidiretto] + Siano $H$ e $K$ due sottogruppi di $G$ con $H \cap K = \{e\}$ e + $H \nsgeq G$. Allora vale che $HK \cong H \rtimes_\varphi K$ con + $\varphi : K \to \Aut(H)$ tale per cui\footnote{ + Tale mappa è ben definita dal momento che $H$ è normale in $G$. + } $k \xmapsto{\varphi} [h \mapsto k h k\inv]$. + \end{theorem} + + \begin{proof} + Si costruisce un isomorfismo tra $H \rtimes_\varphi K$ e $HK$. Sia + $\alpha : H \rtimes_\varphi K \to HK$ tale per cui + $(h, k) \xmapsto{\alpha} hk$. Si verifica che $\alpha$ è un + omomorfismo: + \[ \alpha((h,k)(h',k')) = \alpha(h k h' k\inv, k k') = + h k h' k\inv k k' = hkh'k' = \alpha(h,k)\alpha(h',k'). \] + Chiaramente $\alpha$ è iniettivo dal momento che $hk=e \implies h = k\inv \in H \cap K \implies h = k = e$. Infine $\alpha$ è surgettiva dal momento che $hk = \alpha(h, k)$, + e quindi $\alpha$ è un isomorfismo. + \end{proof} +\end{document} \ No newline at end of file