From 81085a87bc96f4f7bd231fb019baa2563957cd8f Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 13 Nov 2023 12:40:55 +0100 Subject: [PATCH] feat(algebra1): aggiunge risultati immediati tramite Sylow --- .../2. Azione di coniugio e p-gruppi/main.pdf | Bin 201847 -> 203994 bytes .../2. Azione di coniugio e p-gruppi/main.tex | 25 ++++++- .../5. I teoremi di Sylow/main.pdf | Bin 210699 -> 217821 bytes .../5. I teoremi di Sylow/main.tex | 61 +++++++++++++++++- 4 files changed, 83 insertions(+), 3 deletions(-) diff --git a/Secondo anno/Algebra 1/Teoria dei gruppi/Azioni di gruppo e applicazioni/2. Azione di coniugio e p-gruppi/main.pdf b/Secondo anno/Algebra 1/Teoria dei gruppi/Azioni di gruppo e applicazioni/2. 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I teoremi di Sylow/main.tex b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo)/5. I teoremi di Sylow/main.tex index b5a3e74..56f91c6 100644 --- a/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo)/5. I teoremi di Sylow/main.tex +++ b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo)/5. I teoremi di Sylow/main.tex @@ -91,7 +91,32 @@ sicuramente iniettiva (infatti $gx = hx \implies g = h$), e quindi $\abs{\Stab(X)} \leq \abs{X} = p^i$. Si deduce dunque che $\abs{\Stab(X)} = p^i$, da cui la tesi. - \end{proof} \bigskip + \end{proof} + + + \begin{example} + Sia $G$ un $p$-gruppo di ordine $p^n$ con $n \geq 4$ + tale per cui + $\abs{Z(G)} = p$. Si dimostra allora che $G$ ammette un sottogruppo abeliano di ordine $p^3$. \medskip + + + Per il Primo teorema di Sylow, in ogni tale gruppo $G$ + si può estrarre un $p$-sottogruppo $H$ di ordine $p^4$. + Pertanto è sufficiente dimostrare la tesi per $n = 4$, + dacché un sottogruppo di $H$ è in particolare un sottogruppo + di $G$. \medskip + + + Poiché $G$ è un $p$-gruppo tale per cui $\abs{Z(G)} = p$, + esiste $x \in G$ tale per cui $Z_G(x)$ ha ordine $p^3$. + Chiaramente $Z(G) \leq Z(Z_G(x))$, e analogamente + $\gen{x} \leq Z(Z_G(x))$. Pertanto $\abs{Z(Z_G(x))}$ ha + almeno $p^2$ elementi. Se però valesse + $\abs{Z(Z_G(x))} = p^2$, $Z_G(x) \quot Z(Z_G(x))$ sarebbe + ciclico, e quindi $Z_G(x)$ abeliano, \Lightning. Quindi + $Z(Z_G(x))$ ha ordine $p^3$ e coincide con $Z_G(x)$, da + cui la tesi. + \end{example} \bigskip Si dimostra adesso il Secondo teorema di Sylow, che mostra @@ -150,7 +175,17 @@ $\pi_H\inv(\gen{xH})$ è un sottogruppo di $N_P(H)$ di ordine $p \cdot p^i = p^{i+1}$ che contiene $H$, da cui la tesi. - \end{proof} \bigskip + \end{proof} + + \begin{remark} + In particolare, se $G$ è un gruppo abeliano finito, per il + Secondo teorema di Sylow vale che $G(p)$, la $p$-componente + di $G$, è unica in quanto $p$-Sylow di un gruppo abeliano + (infatti l'insieme dei coniugati di un sottogruppo in un gruppo abeliano è sempre banale). Allora $G$ è esattamente + il prodotto diretto dei suoi $p$-Sylow. + \end{remark} + + \bigskip Si dimostra infine il Terzo teorema di Sylow, che riguarda @@ -210,4 +245,26 @@ le orbite sono non banali, e quindi $\abs{\Stab(H)} \neq p^n$), deve dunque valere $n_p \equiv 1 \pod 1$, da cui la tesi. \end{proof} + + \begin{example} + Si mostra che in un gruppo $G$ di ordine $5 \cdot 11 \cdot 17$ + esiste un elemento di ordine $11 \cdot 17$. \medskip + + Si consideri un $11$-Sylow $P_{11}$ e un $17$-Sylow $P_{17}$. + Questi sottogruppi hanno ordine $11$ e $17$, e quindi + sono ciclici. Pertanto esistono $x$, $y \in G$ tali per + cui $P_{11} = \gen{x}$ e $P_{17} = \gen{y}$. \medskip + + Si consideri $n_{11}$: $n_{11}$ deve dividere $5 \cdot 17$, + e quindi $n_{11} \in \{1, 5, 17, 5 \cdot 17\}$. Tuttavia + $n_{11} \equiv 1 \pod{11}$ solo se $n_{11} = 1$. Quindi + $P_{11}$ è l'unico $11$-Sylow, e pertanto è caratteristico, + e dunque normale. Analogamente si verifica che $n_{17} = 1$, + e quindi che anche $P_{17}$ è normale. \medskip + + Poiché $P_{11}$ e $P_{17}$ sono $p$-gruppi relativi a primi + distinti, la loro intersezione è banale. Pertanto + $x$ e $y$ commutano, e allora $\ord(xy) = \ord(x) \ord(y) = + 11 \cdot 17$. + \end{example} \end{document} \ No newline at end of file