From 3688da4b323d5f3fa8662843535429cf38b60675 Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 19 Jul 2023 18:53:14 +0200 Subject: [PATCH] fix(algebra1): elimina parte di un enunciato ridondante --- .../3. Azione di coniugio e p-gruppi/main.pdf | Bin 157574 -> 157526 bytes .../3. Azione di coniugio e p-gruppi/main.tex | 2 +- .../main.pdf | Bin 197622 -> 209587 bytes .../main.tex | 9 ++++++++- 4 files changed, 9 insertions(+), 2 deletions(-) diff --git a/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.pdf b/Secondo anno/Algebra 1/3. 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Azione di coniugio e p-gruppi/main.tex b/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex index cd46e95..d9d88fc 100644 --- a/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex +++ b/Secondo anno/Algebra 1/3. Azione di coniugio e p-gruppi/main.tex @@ -54,7 +54,7 @@ Infatti, grazie alla formula delle classi di coniugio, si osserva facilmente che il centro di un $p$-gruppo non è mai banale (ossia composto dalla sola identità), come mostra la: \begin{proposition} - Sia $G$ un $p$-gruppo. Allora $\abs{Z(G)} = pk$, con $k \in \NN^+$, $k \geq 1$. + Sia $G$ un $p$-gruppo. Allora $\abs{Z(G)} > 1$. \end{proposition} \begin{proof} diff --git a/Secondo anno/Algebra 1/6. Prodotto di sottogruppi e ordini di gruppi abeliani/main.pdf b/Secondo anno/Algebra 1/6. 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Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex b/Secondo anno/Algebra 1/6. Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex index 99ce968..838db45 100644 --- a/Secondo anno/Algebra 1/6. Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex +++ b/Secondo anno/Algebra 1/6. Prodotto di sottogruppi e ordini di gruppi abeliani/main.tex @@ -153,7 +153,14 @@ Chiaramente $\rho$ è surgettiva. Inoltre $\rho((h,k)) = e \implies h = k\inv \in H \cap K$, e dunque $h = k = e$, da cui l'iniettività di $\rho$ e la tesi. - \end{proof} \bigskip + \end{proof} + + + Inoltre, se $G = G_1 \times G_2$ con $G_1$ e $G_2$ gruppi, si possono trovare + facilmente due copie isomorfe di $G_1$ e $G_2$ in $G$, ossia $G_1' = G_1 \times \{e\}$ e $G_2' = \{e\} \times G_2$. + Vale inoltre che $G_1'$, $G_2' \nsgeq G$ e dunque, + per la proposizione precedente\footnote{Infatti $G_1' \cap G_2' = \{(e,e)\}$.}, che + $G \cong G_1' \times G_2'$. \bigskip Si può adesso dimostrare il seguente fondamentale