From fb44cc994da9aa7ab46a710ad3a1687dc98a9678 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 1 Oct 2023 13:24:35 +0200 Subject: [PATCH] feat(algebra1): aggiunge note su D_n e sui sottogruppi normali --- .../main.pdf | Bin 193431 -> 197662 bytes .../main.tex | 35 ++++- .../main.pdf | Bin 0 -> 241466 bytes .../main.tex | 146 ++++++++++++++++++ tex/latex/style/notes_2023.sty | 8 + 5 files changed, 188 insertions(+), 1 deletion(-) create mode 100644 Secondo anno/Algebra 1/8. Il gruppo diedrale e i suoi sottogruppi/main.pdf create mode 100644 Secondo anno/Algebra 1/8. Il gruppo diedrale e i suoi sottogruppi/main.tex diff --git a/Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.pdf b/Secondo anno/Algebra 1/5. 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Il gruppo diedrale e i suoi sottogruppi/main.tex b/Secondo anno/Algebra 1/8. Il gruppo diedrale e i suoi sottogruppi/main.tex new file mode 100644 index 0000000..91e853c --- /dev/null +++ b/Secondo anno/Algebra 1/8. Il gruppo diedrale e i suoi sottogruppi/main.tex @@ -0,0 +1,146 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Il gruppo diedrale e i suoi sottogruppi} + \maketitle + + In questo documento si definisce il gruppo diedrale e si illustrano + le sue proprietà principali, a partire da come sono costruiti i suoi + sottogruppi. \medskip + + + Sia $n \geq 3$. Si definisce \textbf{gruppo diedrale}, denotato\footnote{ + Alcuni testi denotano il gruppo diedrale come $D_{2n}$, dal + momento che vale $\abs{D_n} = 2n$. + } come $D_n$, il gruppo delle isometrie del piano $\RR^2$ che mappano i vertici di + un poligono regolare centrato nell'origine con $n$ lati in sé stessi. \medskip + + + Si verifica facilmente che $D_n$ è un gruppo: + \begin{itemize} + \item Ammette un'identità, che coincide con l'identità delle isometrie, + \item La composizione di due isometrie che mappano i vertici del poligono in + sé stessi è ancora un'isometria che lascia fissi i vertici del poligono, + \item Ogni isometria per cui i vertici del poligono rimangono fissi ammette + un'inversa con la stessa proprietà\footnote{ + Si ricorda che ogni isometria è invertibile a prescindere. + }\footnote{ + Dal momento che $D_n$ ha cardinalità $2n$, come mostrato dopo, questa + condizione è automaticamente verificata come conseguenza della finitezza + di $D_n$. + }. + \end{itemize} + + In particolare, se $\sigma \in D_n$, $\sigma$ permuta i vertici del poligono (pertanto + si può visualizzare $D_n$ come un sottogruppo naturale di $S_n$). Denotando con + $r$ la rotazione primitiva del gruppo (ossia una rotazione di $\frac{2\pi}{n}$ gradi in + senso antiorario) e con $s$ la simmetria rispetto all'asse $y$, si osserva che + ogni elemento della forma $s r^k$ con $k \in \ZZ$ è ancora una simmetria, benché non + per forza rispetto all'asse $y$\footnote{ + La matrice associata di $s$ nella base canonica è $-1 E_{11} + E_{22}$, e quindi deve valere $\det(s) = -1$. Al contrario $r \in \SOO(2)$, e quindi $\det(r) = 1$. Si conclude pertanto che + $\det(s r^k) = \det(s) \det(r)^k = -1$, e dunque che $s r^k$ deve obbligatoriamente + appartenere alla classe laterale $s \SOO(2)$ delle riflessioni. + }. In particolare, per $n$ pari, le riflessioni di $D_n$ sono esattamente le riflessioni + rispetto alle rette passanti per i vertici o per i punti medi del poligono. \medskip + + + Dal momento che $\sigma \in D_n$ è in particolare una isometria, e quindi + un'applicazione lineare, $\sigma$ è completamente determinata da + $\sigma(V_1)$ e $\sigma(V_2)$, dove $V_i$ sono i vertici del poligono numerati + in senso antiorario. In particolare, se $\sigma(V_1) = V_k$, allora + $\sigma(V_2)$, affinché venga preservata la distanza, può valere\footnote{ + Per semplicità si pone $V_0 := V_n$ e $V_{n+1} := V_1$. + } o + $V_{k-1}$ o $V_{k+1}$. Pertanto vi sono al più $2n$ scelte possibili di + $\sigma(V_1)$ e $\sigma(V_2)$ (e quindi $\abs{D_n} \leq 2n$). D'altra parte + si osserva che tutti gli elementi $1$, $r$, ..., $r^{n-1}$, $s$, $sr$, ..., $s r^{n-1}$ + sono distinti: + \begin{itemize} + \item Gli $r^k$ con $0 \leq k \leq \ord(r) - 1$ sono tutti distinti e $\ord(r)$ vale + esattamente\footnote{ + Infatti $r$ è rappresentato in $\SOO(2)$ dalla matrice $\SMatrix{ + \cos(\frac{2\pi}{n}) & -\sin(\frac{2\pi}{n}) \\ + \sin(\frac{2\pi}{n}) & \cos(\frac{2\pi}{n}) + }$, che ha ordine esattamente $n$. + } $n$, + \item Gli $sr^k$ con $0 \leq k \leq n - 1$ sono tutti distinti, altrimenti + la precedente osservazione sarebbe contraddetta, + \item Nessun $r^i$ coincide con un $s r^j$, dal momento che i loro determinanti + sono diversi ($\det(r^i) = 1$, mentre $\det(s r^j) = -1$). In particolare + $r^i \in \SOO(2)$, mentre $s r^j \in s \SOO(2)$. + \end{itemize} + + Pertanto $\abs{D_n} \geq 2n$, e quindi $\abs{D_n} = 2n$. Si conclude inoltre + che $D_n$ è generato da $r$ e da $s$, e quindi che $D_n = \gen{r, s}$. Esistono + dunque due sottogruppi naturali di $D_n$: + \[ \rotations := \gen{r} \cong \ZZ \quot n\ZZ, \quad \gen{s} \cong \ZZ \quot 2\ZZ. \] + + + \begin{proposition} + Vale l'identità $s r s\inv = r\inv$. + \end{proposition} + + \begin{proof} + Si sviluppa $s r s\inv$ in termini matriciali, considerando + $s = \SMatrix{ + -1 & 0 \\ + 0 & 1 + }$ e $r = \SMatrix{ + \cos(\frac{2\pi}{n}) & -\sin(\frac{2\pi}{n}) \\ + \sin(\frac{2\pi}{n}) & \cos(\frac{2\pi}{n}) + }$: + \[ s r s\inv = \Matrix{ + \cos(\frac{2\pi}{n}) & \sin(\frac{2\pi}{n}) \\ + -\sin(\frac{2\pi}{n}) & \cos(\frac{2\pi}{n}) + }, \] + ottenendo la matrice associata a $r\inv$ nella base canonica. + \end{proof} + + + In generale vale dunque che $s r^k s\inv = r^{-k}$. Si deduce allora la presentazione del gruppo $D_n$: + \[ D_n = \gen{r,s \mid r^n = 1, s^2 = 1, s r s\inv = r\inv}. \] \smallskip + + + Si descrivono adesso tutti i sottogruppi di $D_n$. Innanzitutto, in $\rotations$ + per ogni $d \mid n$ esiste un unico sottogruppo di ordine $d$ dal momento che + $\rotations$ è ciclico. Pertanto ogni tale sottogruppo assume la forma + $\gen{r^{\frac{n}{d}}}$. Inoltre, dal momento che\footnote{ + Infatti ogni elemento di $D_n$, come visto prima, è della $r^k$ o $s r^k$. + } $[D_n : \rotations] = 2$, $\rotations$ è un sottogruppo normale di + $D_n$. Allora, poiché $\rotations$ è normale in $D_n$ e ogni sottogruppo + $H \leq \rotations$ è caratteristico\footnote{ + Per ogni ordine di $\rotations$ esiste un unico sottogruppo $H \leq \rotations$, + e quindi tale sottogruppo deve essere caratteristico. + } in $D_n$, ogni sottogruppo di $\rotations$ è normale anche in $D_n$. \medskip + + + Sia ora $H$ un sottogruppo di $D_n$ con $H \not\subseteq \rotations$. Si consideri + la proiezione al quoziente mediante $\rotations$, ossia $\pi_\rotations : D_n \to D_n / + \rotations$. Chiaramente deve valere che $\pi_\rotations(H) = D_n / \rotations$: l'unica + altra possibilità è che $\pi_\rotations(H)$ sia $\{\rotations\}$, e quindi che + $H \subseteq \Ker \pi_\rotations = \rotations$, \Lightning. \medskip + + + Si consideri adesso + la restrizione di $\pi_\rotations$ ad $H$, $\restr{\pi_\rotations}{H} : H \to D_n / \rotations$. Vale in particolare che $\Ker \restr{\pi_\rotations}{H} = H \cap \Ker \pi_\rotations = H \cap \rotations$ e che $\Im \restr{\pi_\rotations}{H} = D_n / \rotations$ + (da prima vale infatti che $\pi_\rotations(H) = D_n / \rotations$). Allora, per il Primo + teorema di isomorfismo, vale che: + \[ \frac{H}{H \cap \rotations} \cong D_n \quot \rotations, \] + da cui si deduce che $\abs{H} = 2 \abs{H \cap \rotations}$. In particolare $H \cap \rotations$ è un sottogruppo di $\rotations$, e quindi esiste $d \mid n$ tale per cui + $H \cap \rotations = \gen{r^d}$, con $\abs{H \cap \rotations} = \frac{n}{d}$. \medskip + + + Sia ora $s r^k$ una simmetria di $H$. + Innanzitutto si osserva che $\gen{r^d}$ è normale in $D_n$ e quindi + $\gen{ r^d }\gen{ s r^k }$ è effettivamente un sottogruppo di $D_n$. Dal momento che\footnote{ + Infatti l'unica rotazione che è anche una simmetria è l'identità. + } + $\gen{ r^d } \cap \gen{ s r^k } = \{ e \}$, allora $\abs{\gen{ r^d } \gen{ s r^k }} = \abs{\gen{r^d}} \abs{\gen{s r^k}} = \frac{2n}{d}$. Anche $\abs{H} = \frac{2n}{d}$ e + quindi, per questioni di cardinalità, $H = \gen{ r^d } \gen{ s r^k } = \gen{ r^d, s r^k }$. + \medskip + + + In conclusione, ogni sottogruppo di $D_n$ è della forma $\gen{r^d}$ o della forma + $\gen{r^d, sr^k}$. +\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 ef20743..ec59ed9 100644 --- a/tex/latex/style/notes_2023.sty +++ b/tex/latex/style/notes_2023.sty @@ -18,6 +18,8 @@ \usepackage{stackengine} \usepackage{wasysym} +\usepackage{tikz-cd} + \usepackage[italian]{babel} \usepackage{tabularx} @@ -106,6 +108,10 @@ \newcommand{\RRbar}{\overline{\RR}} % Spesso utilizzati al corso di Geometria 1. + +\DeclareMathOperator{\OO}{O} % gruppo ortogonale +\DeclareMathOperator{\SOO}{SO} % gruppo ortogonale speciale + \newcommand{\proj}[1]{\Matrix{#1 \\[0.03in] \hline 1}} \newcommand{\projT}[1]{\Matrix{#1 & \rvline & 1}^\top} @@ -209,6 +215,8 @@ % Spesso utilizzati durante il corso di Algebra 1 +\newcommand{\rotations}{\mathcal{R}} +\newcommand{\gen}[1]{\langle #1 \rangle} \DeclareMathOperator{\Cl}{Cl} \newcommand{\actson}{\circlearrowleft}