From 797959018309b83be829640d8cd6ca6d9c119880 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 20 Oct 2023 12:39:31 +0200 Subject: [PATCH] feat(algebra1): aggiunge la teoria sulle azioni naturali e sui sottogruppi transitivi di S_n --- .../10. Il gruppo delle permutazioni/main.pdf | Bin 210749 -> 246728 bytes .../10. Il gruppo delle permutazioni/main.tex | 83 +++++++++++++++++- 2 files changed, 80 insertions(+), 3 deletions(-) diff --git a/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.pdf b/Secondo anno/Algebra 1/10. 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Il gruppo delle permutazioni/main.tex b/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex index b789f44..eb0b2ac 100644 --- a/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex +++ b/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex @@ -10,19 +10,62 @@ $\{1, \ldots, n\}$ e con $G$ un qualsiasi gruppo. \end{note} - Si definisce brevemente il gruppo delle permutazioni $S_n$ come il gruppo + Si definisce brevemente il \textbf{gruppo delle permutazioni} $S_n$ come il gruppo delle bigezioni su $G$, ossia $S(X_n)$. Si deduce facilmente che $\abs{S_n} = n!$ dal momento che vi sono esattamente $n!$ scelte possibili per costruire una bigezione da $X_n$ in $X_n$ stesso. \medskip + Si definisce + l'\textbf{azione naturale} di $S_n$ su $X_n$ come l'azione $\varphi : S_n \to S(X_n)$ + tale per cui $\sigma \xmapsto{\varphi} [n \mapsto \sigma(n)]$. In particolare, + per $H \leq S_n$, si definisce la sua azione naturale come la restrizione dell'azione + naturale di $S_n$ su $H$. Un sottogruppo $H$ si dice \textit{transitivo} se la + sua azione naturale è transitiva. Si osserva che ogni tale azione naturale è fedele + (infatti $\sigma \in S_n$ fissa tutto $X_n$ solo se è l'identità di $S_n$). Si illustra allora subito un risultato sui + sottogruppi abeliani transitivi di $S_n$: + + \begin{proposition} + Sia $H$ un sottogruppo abeliano transitivo di $S_n$. Allora + $\abs{H} = n$. + \end{proposition} + + \begin{proof} + Dal Teorema orbita-stabilizzatore, $\abs{H} = \abs{\Stab(i)} \abs{\Orb(i)}$. Poiché + $H$ è un sottogruppo transitivo, $\abs{\Orb(i)} = n$, e quindi è sufficiente + verificare che $\Stab(i)$ sia banale. \medskip + + + Ogni $\Stab(i)$ è coniugato + ad ogni altro $\Stab(j)$, sempre per la transitività dell'azione; poiché allora $H$ è abeliano, in particolare $\Stab(i)$ coincide con + ogni altro stabilizzatore. Pertanto $\sigma \in \Stab(i)$ se e solo se + $\sigma$ appartiene al nucleo dell'azione naturale di $H$, ossia + a $\cap_{x=1}^n \Stab(x)$, e quindi se e solo se $\sigma = e$. Si conclude dunque + che $\Stab(i)$ è banale e quindi che $\abs{H} = n$. + \end{proof} + + \begin{example}[Il gruppo di Klein $V_4$] + In $S_4$, e in particolare in $A_4$, esiste un sottogruppo normale non banale + molto particolare\footnote{ + Pertanto $A_4$ non è semplice. + }, il cosiddetto\footnote{ + La lettere $V$ è dovuta al termine \textit{vier}, che in tedesco + significa ``quattro''. + } \textbf{gruppo di Klein} $V_4$, dove: + \[ V_4 = \{ e, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3) \}. \] + Tale sottogruppo è abeliano e transitivo (e quindi, per il risultato di prima, + $\abs{V_4} = 4$, come si osserva facilmente). Poiché ogni suo elemento ha + ordine $2$ (e in particolare $V_4$ non è ciclico), $V_4$ deve necessariamente + essere isomorfo a $\ZZmod2 \times \ZZmod2$. Pertanto $V_4$ è il più piccolo + gruppo non ciclico per ordine (a meno di isomorfismo). + \end{example} + Come è noto, ogni $\sigma \in S_n$ può scriversi come prodotto di cicli disgiunti. Di seguito si introduce un modo formale per descrivere questi cicli. \medskip - Si consideri l'azione di $\gen{\sigma}$ su $X_n$ univocamente determinata - da $\sigma \cdot x = \sigma(x)$. Allora i cicli di $\sigma$ sono esattamente + Si consideri l'azione naturale di $\gen{\sigma}$. Allora i cicli di $\sigma$ sono esattamente le orbite di $\sigma$ ordinate nel seguente modo: \[ \Orb(x) = \{ x, \sigma(x), \dots, \sigma^m(x) \}. \] @@ -108,4 +151,38 @@ se $H \leq S_n$, $H$ è normale in $S_n$ se e solo se per ogni tipo di permutazione $H$ contiene tutte le permutazioni di quel tipo o nessuna. \medskip + + + Si osserva adesso che $\An$ può scriversi come il sottogruppo generato dai + $2-2$-cicli, infatti ogni permutazione pari è prodotto di un numero pari di trasposizioni, + che possono dunque essere ridotte a $2-2$-cicli. Allo stesso tempo allora + $\An$ è generato dai $3$-cicli se $n \geq 3$. Si consideri infatti $(i, j)(k, l)$. Se + $\{i, j\} \cap \{k, l\} = 2$, $(i, j) = (k, l)$, e quindi $(i, j)(k, l) = e$; + se $\{i, j\} \cap \{k, l\} = 1$, si può assumere senza perdita di generalità che + $k = i$, da cui $(i, j)(i, l) = (i, l, j)$, un $3$-ciclo; se invece $\{i, j\} \cap \{k, l\} = 0$, + $(i, j)(k, l) = (i, j)(j, k)(j, k)(k, l) = (i, j, k)(j, k, l)$, e quindi + $(i, j)(k, l)$ è prodotto di due $3$-cicli. Pertanto si è dimostrato + che $\An = \gen{(i, j)(k, l) \mid i, j, k, l \in X_n} \subseteq \gen{(i, j, k) \mid i, j, k \in X_n}$; allo stesso tempo ogni $3$-ciclo è una permutazione + pari, e quindi vale anche l'inclusione inversa. \medskip + + + + Si consideri adesso $S_n'$, il sottogruppo derivato di $S_n$. Poiché $S_n$ è abeliano + per $n \in \{ 1, 2 \}$, in tal caso $S_n' = \{e\}$; in tutti gli altri casi + $S_n'$ non può essere uguale a $\{e\}$, altrimenti $S_n$ sarebbe abeliano. Si osserva + che $[(i,j), (j,k)]$ con $i$, $j$ e $k$ distinti si scrive come: + \[ [(i,j), (j,k)] = (i, j) (j, k) (i, j)\inv (j, k)\inv = (i, k) (j, k) = (i, k, j), \] + e quindi si deduce che $\gen{(i, j, k) \mid \abs{\{i, j, k\}} = 3} = \An$ è un sottogruppo + di $S_n'$. Inoltre\footnote{ + Alternativamente $[S_n : S_n']$ deve dividere $[S_n : \An] = 2$, e quindi, + poiché $S_n \neq S_n'$, è necessario che $S_n'$ sia esattamente $\An$. + } l'omomorfismo $\sgn$ ha come codominio un gruppo abeliano isomorfo + a $\ZZmod2$, e quindi $S_n' \subseteq \Ker \sgn = \An$. Si conclude dunque che + $S_n' = \An$ e che ${S_n}_{\,ab} = S_n \quot \An \cong \{\pm 1\} \cong \ZZmod2$ per $n \geq 3$. Pertanto adesso è immediato il seguente risultato: + + \begin{proposition} + Sia $H$ un gruppo abeliano. Allora $\Hom(S_n, H) \bij \Hom(\ZZmod2, H)$. + \end{proposition} + + In particolare, vi sono tanti omomorfismi non banali in $\Hom(S_n, H) \bij \Hom(\ZZmod2, H)$ quanti elementi di ordine $2$ vi sono in $H$. \end{document} \ No newline at end of file