From 05a086ca7d28c9b25fa6b414a1e1099a00828452 Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 16 Oct 2023 20:17:17 +0200 Subject: [PATCH] =?UTF-8?q?feat(algebra1):=20aggiunge=20il=20teorema=20di?= =?UTF-8?q?=20Poincar=C3=A9=20e=20i=20risultati=20collegati?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../main.pdf | Bin 197662 -> 210616 bytes .../main.tex | 72 ++++++++++++++++++ 2 files changed, 72 insertions(+) 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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Normalizzatore e teorema di Cayley/main.tex b/Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.tex index a259496..c77f5c8 100644 --- a/Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.tex +++ b/Secondo anno/Algebra 1/5. Normalizzatore e teorema di Cayley/main.tex @@ -77,4 +77,76 @@ poiché $K$ è caratteristico in $H$, $\restr{\varphi_g}{H}(K) = K$, da cui si deduce che $gKg\inv = K$ per ogni $g \in G$. \end{proof} + + Si illustra adesso un risultato riguardante l'esistenza di sottogruppi normali in $G$: + \begin{theorem}[di Poincaré] + Sia $H$ un sottogruppo di $G$ di indice $n$. Allora esiste sempre un sottogruppo + $N$ di $G$ tale per cui: + \begin{enumerate}[(i)] + \item $N$ è normale in $G$, + \item $N$ è contenuto in $H$, + \item $n \mid [G : N] \mid n!$. + \end{enumerate} + \end{theorem} + + \begin{proof} + Si consideri l'azione $\varphi : G \to S(G \quot H)$ tale per cui + $g \xmapsto{\varphi} [kH \mapsto gkK]$. Tale azione è sicuramente + ben definita dal momento che $kH = k'H \implies gkH = gk'H$. Si + studia $N := \Ker \varphi$. Chiaramente $N$ è normale in $G$, e si + verifica facilmente che $N$ è contenuto anche in $H$, infatti, se + $n \in N$, allora: + \[ H = \varphi(n)(H) = nH \implies n \in H. \] + Poiché $G \quot N$ è isomorfo a $\Im \varphi \leq S(G \quot H)$, + $[G : N] \mid \abs{S(G \quot H)} = \abs{S_n} = n!$ considerando che + $S(G \quot H) \cong S_n$. Dal momento allora che $N$ è un sottogruppo + di $H$, vale che: + \[ [G : N] = [G : H] [H : N] = n [H : N], \] + e quindi $n \mid [G : N]$. Si è dunque esibito un sottogruppo $N$ con + le proprietà indicate nella tesi. + \end{proof} + + Dal precedente teorema sono immediati i seguenti due risultati: + + \begin{corollary} + Sia $H$ un sottogruppo di $G$ con indice $n$. Se $n! < \abs{G}$ e + $n>1$, allora $G$ non è semplice. + \end{corollary} + + \begin{corollary} + Sia $H$ un sottogruppo di $G$ con indice $p$, dove $p$ è il più piccolo + primo che divide $n = \abs{G}$. Allora $H$ è normale. + \end{corollary} + + \begin{proof} + Per il Teorema di Poincaré, esiste un sottogruppo $N$ di $H$ tale per cui + $N$ sia normale e $p \mid [G : N] \mid p!$ con $p = [G : H]$. In particolare + $[G : N]$ deve dividere anche $n$, e quindi $[G : N]$ deve dunque + dividere $\MCD(p!, n)$, che è, per ipotesi, $p$ stesso. Si conclude dunque + che $[G : N] = p = [G : H]$, e quindi che $N = H$, ossia che $H$ stesso + è normale. + \end{proof} + + \begin{example} [Tutti i gruppi di ordine $15$ sono ciclici] + Sia $G$ un gruppo di ordine $15$. Per il teorema di Cauchy esistono + due elementi $h$ ed $k$, uno di ordine $3$ e l'altro di ordine $5$. + In particolare, si consideri $K = \gen{k}$; poiché $\abs{K} = 5$, + $[G : K] = 3$, il più piccolo primo che divide $15$. Pertanto + $K$ è normale per il corollario di sopra. \medskip + + + Poiché $K$ è normale, si può considerare la restrizione $\iota : + \Inn(G) \to \Aut(K)$ tale per cui $\varphi_g \xmapsto{\iota} \restr{\varphi_g}{K}$. + Dal momento che $K$ è ciclico, $\Aut(K) \cong \Aut(\ZZ \quot 5 \ZZ) \cong + (\ZZ \quot 5 \ZZ)^* \cong \ZZ \quot 4 \ZZ$. Quindi $[G : \Ker \iota]$ deve + dividere sia $4$ che $15$; dal momento che $\MCD(4, 15) = 1$, $[G : \Ker \iota] = 1$, + e quindi che $\iota$ è l'omomorfismo banale. Poiché $\iota$ è banale, $K$ è + un sottogruppo di $Z(G)$. \medskip + + + In particolare $[G : Z(G)] \mid [G : K] = 3$, e quindi in particolare + $G \quot Z(G)$ è ciclico, da cui si deduce che $G$ è abeliano. Infine, + dal momento che $\MCD(3, 5) = 1$ e $h$ e $k$ commutano, + $hk$ è un elemento di ordine $15$, e dunque $G$ è ciclico. + \end{example} \end{document} \ No newline at end of file