From 9e8ac5ee0a78c537259d8459e616e303aa89e26f Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 15 Nov 2023 10:06:22 +0100 Subject: [PATCH] fix(algebra1): elimina un typo --- .../main.pdf | Bin 224116 -> 228780 bytes .../main.tex | 18 +++++++++++++++--- 2 files changed, 15 insertions(+), 3 deletions(-) diff --git a/Secondo anno/Algebra 1/Teoria dei gruppi/Costruzioni di gruppo (prodotti di sottogruppi, semidiretti e presentazioni)/1. Prodotto di sottogruppi e ordini di gruppi abeliani/main.pdf b/Secondo anno/Algebra 1/Teoria dei gruppi/Costruzioni di gruppo (prodotti di sottogruppi, semidiretti e presentazioni)/1. 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Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex b/Secondo anno/Algebra 1/Teoria dei gruppi/Costruzioni di gruppo (prodotti di sottogruppi, semidiretti e presentazioni)/1. Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex index df21688..c8e7c63 100644 --- a/Secondo anno/Algebra 1/Teoria dei gruppi/Costruzioni di gruppo (prodotti di sottogruppi, semidiretti e presentazioni)/1. Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex +++ b/Secondo anno/Algebra 1/Teoria dei gruppi/Costruzioni di gruppo (prodotti di sottogruppi, semidiretti e presentazioni)/1. Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex @@ -196,7 +196,7 @@ 1$, e quindi $\gen{x} \cap \gen{y} = \{e\}$. Allora deve valere che $x^k = y^{-k} = e \implies \ord(x), \ord(y) \mid k$, da cui si deduce che - $\ord(x) \ord(y) \mid k = \ord(x) \ord(y)$. Si conclude dunque che + $\ord(x) \ord(y) \mid k = \ord(xy)$. Si conclude dunque che $\ord(xy) = \ord(x) \ord(y)$. \end{proof} @@ -241,13 +241,25 @@ teorema per i gruppi abeliani: \begin{theorem} - Sia $G$ un gruppo abeliano finito di ordine $n$. + Sia\footnote{ + In realtà questo teorema diventa di facile + dimostrazione una + volta che si dimostra il Teorema di struttura + per gruppi abeliani finiti. È sufficiente + infatti dividere $G$ nel prodotto delle sue + $p$-componenti ed estrarre da ogni $p$-componente + un sottogruppo affinché il prodotto dei sottogruppi + scelti abbia ordine $m$. + } $G$ un gruppo abeliano finito di ordine $n$. Allora, se $m$ divide $n$, esiste un sottogruppo di $G$ di ordine $m$. \end{theorem} \begin{proof} - Si dimostra preliminarmente che se $p^k$ divide $n$, + Si\footnote{ + Questa parte della dimostrazione è già implicata + dal Primo teorema di Sylow. + } dimostra preliminarmente che se $p^k$ divide $n$, dove $p$ è un numero primo e $k \in \NN^+$, allora $G$ ammette un sottogruppo di ordine $p^k$. Si mostra la tesi per induzione su $k$. \medskip