From ea8ccc81f433cf88f6ebfebefb656cbeb183fab4 Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 13 Nov 2023 12:13:32 +0100 Subject: [PATCH] feat(algebra1): aggiunge i teoremi di Sylow --- .../2. Azione di coniugio e p-gruppi/main.pdf | Bin 197017 -> 201847 bytes .../2. Azione di coniugio e p-gruppi/main.tex | 39 ++++ .../5. I teoremi di Sylow/main.pdf | Bin 0 -> 210699 bytes .../5. I teoremi di Sylow/main.tex | 213 ++++++++++++++++++ tex/latex/style/notes_2023.sty | 7 + 5 files changed, 259 insertions(+) create mode 100644 Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo)/5. I teoremi di Sylow/main.pdf create mode 100644 Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo)/5. I teoremi di Sylow/main.tex 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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\setminus H$, e quindi un elemento $x$ + appartenente a $N_G(H)$, ma non ad $H$. In tal caso, + si deduce facilmente che $H \lneq N_G(H)$. \medskip + + + Se invece $Z(G) \leq H$, si può applicare il Teorema + di corrispondenza. Poiché $G \quot Z(G)$ è un $p$-gruppo + di ordine strettamente minore di $p^n$ (infatti il + centro di un $p$-gruppo è sempre non banale), per + induzione $H \quot Z(G) \lneq N_{G \quot Z(G)}(H \quot Z(G))$. + Allora, per il Teorema di corrispondenza, + $H = \pi_{Z(G)}\inv(H \quot Z(G)) \lneq \pi_{Z(G)}\inv( + N_{G \quot Z(G)}(H \quot Z(G)))$. È sufficiente mostrare + che $\pi_{Z(G)}\inv( + N_{G \quot Z(G)}(H \quot Z(G))) \subseteq N_G(H)$ per + dedurre la tesi. Sia allora $g \in \pi_{Z(G)}\inv( + N_{G \quot Z(G)}(H \quot Z(G)))$. Allora, per ipotesi, + vale che: + \[ \pi_{Z(G)}(gHg\inv) = g Z(G) \pi_{Z(G)}(H) g\inv Z(G) \subseteq \pi_{Z(G)}(H), \] + per cui $gHg\inv \subseteq H$. + \end{proof} \end{document} \ No newline at end of file diff --git a/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo)/5. I teoremi di Sylow/main.pdf b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo)/5. 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