From 50d0d6e913b798f5c651616417659b379dcc5a3b Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 15 Nov 2023 20:38:51 +0100 Subject: [PATCH] feat(algebra1): aggiunge la dimostrazione del primo teorema utilizzato per il teorema di struttura --- .../main.pdf | Bin 248828 -> 250317 bytes .../main.tex | 33 +++++++++++++----- 2 files changed, 25 insertions(+), 8 deletions(-) diff --git a/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.pdf b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. 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Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex index 2f8cb17..ab98ab9 100644 --- a/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex +++ b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex @@ -116,7 +116,7 @@ \item Se $G$ è abeliano con $\abs{G} = p_1^{e_1} \cdots p_r^{e_r}$, allora $G \cong G(p_1) \times \cdots \times G(p_r)$, ossia $G$ è isomorfo al prodotto diretto tra le sue $p$-componenti. Tale decomposizione di $G$ come prodotto di $p$-gruppi di ordini tra loro - coprimi è unica. + coprimi è unica a meno di isomorfismi tra i vari $p$-gruppi. \item Se $G$ è un $p$-gruppo abeliano. Allora esistono e sono univocamente determinati degli interi positivi $r_1 \geq \cdots \geq r_s$ tali che @@ -171,11 +171,29 @@ Se $G$ è abeliano con $\abs{G} = p_1^{e_1} \cdots p_r^{e_r}$, allora $G \cong G(p_1) \times \cdots \times G(p_r)$, ossia $G$ è isomorfo al prodotto diretto tra le sue $p$-componenti. Tale decomposizione di $G$ come prodotto di $p$-gruppi di ordini tra loro - coprimi è unica. + coprimi è unica a meno di isomorfismi tra i vari $p$-gruppi. \end{theorem} \begin{proof} - %TODO + Poiché $G$ è abeliano, esiste il sottogruppo $G(p_1) G(p_2) \cdots G(p_r)$. + In particolare, dal momento che $\MCD(\abs{G(p_i)}, \abs{G(p_j)}) = 1$ e che + $G(p_i) \cap G(p_j) = \{e\}$ per + $i \neq j$, vale anche che $\abs{G(p_1) G(p_2) \cdots G(p_r)} = \abs{G}$. + Allora deve valere in particolare che $G = G(p_1) G(p_2) \cdots G(p_r)$. Per + il Teorema di decomposizione in prodotto diretto, si deduce che + $G \cong G(p_1) \times \cdots \times G(p_r)$. \medskip + + + Si mostra che la decomposizione di $G$ come prodotto di $p$-gruppi di ordini + tra loro coprimi è unica. Sia $H_1 \times H_2 \times \cdots H_r$ un'altra + decomposizione di $G$ in prodotto diretto in modo tale che $\abs{H_i} = p_i^{e_i}$. + Sia $\varphi$ un isomorfismo da $H_1 \times H_2 \times \cdots \times H_r$ in + $G(p_1) \times G(p_2) \times \cdots \times G(p_r)$. Allora $\varphi$ ristretto + all'identificazione di $H_i$ in $H_1 \times H_2 \times \cdots H_r$ (ossia + $\{e\} \times \cdots \times H_i \times \cdots \times \{e\}$) nell'identificazione + di $G(p_i)$ è ancora un isomorfismo, dal momento che entrambi sono gli unici + $p$-gruppi che compaiono nelle rispettive fattorizzazioni. Allora + $H_i \cong G(p_i)$, concludendo la dimostrazione. \end{proof} \begin{theorem} @@ -188,12 +206,11 @@ \begin{proof} %TODO - \end{proof} + \end{proof} \bigskip - I gruppi moltiplicativi $\ZZmulmod{p^k}$ e $\ZZmulmod{2p^k}$, - con $p$ numero primo, sono completamente classificati e - sono note le loro decomposizioni in fattori invarianti, - come mostra il fondamentale: + I gruppi moltiplicativi $\ZZmulmod{n}$ sono completamente classificati ed + è noto l'algoritmo per dedurre le loro decomposizioni in fattori invarianti, + come mostra il fondamentale \begin{theorem} Sia $p$ un numero primo dispari e $k \in \NN^+$. Allora,