From 1eb80ffa1ea046bcb928768476946de4963b7e67 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 1 Oct 2023 18:55:45 +0200 Subject: [PATCH] feat(algebra1): isomorfismi dei gruppi di ordine p^2 --- .../1. Il gruppo degli automorfismi/main.pdf | Bin 205454 -> 205454 bytes .../3. Azione di coniugio e p-gruppi/main.pdf | Bin 157526 -> 197017 bytes .../3. Azione di coniugio e p-gruppi/main.tex | 39 ++++++++++++++++-- 3 files changed, 35 insertions(+), 4 deletions(-) diff --git a/Secondo anno/Algebra 1/1. Il gruppo degli automorfismi/main.pdf b/Secondo anno/Algebra 1/1. 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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 d9d88fc..2c86a0e 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 @@ -87,9 +87,40 @@ dal momento che si era presupposto che $Z(G)$ fosse un sottogruppo proprio di $G$, \Lightning. Allora $Z(G)$ ha ordine $p^2$, e dunque $Z(G) = G$. - \end{proof} + \end{proof} \medskip + - Questo risultato ha un'immediata applicazione, se combinato con il Teorema - fondamentale dei gruppi abeliani finiti. Infatti, esso implica che - $G$ deve per forza essere isomorfo a $\ZZ_{p^2}$ o a $\ZZ_{p} \times \ZZ_{p}$. + \begin{example} + Si mostra che\footnote{ + Il risultato è facilmente dimostrabile attraverso + il Teorema di struttura dei gruppi abeliani + finitamente generati. + } $G$ è obbligatoriamente isomorfo a + $\ZZ_{p^2}$ o a $\ZZ_p \times \ZZ_p$ se + $\abs{G} = p^2$. \vskip 0.1in + + + Se $G$ ammette un generatore, + allora $G$ è ciclico e quindi isomorfo a $\ZZ_{p^2}$. + Altrimenti, sia $g \in G$ un elemento di ordine\footnote{ + Questo elemento deve esistere obbligatoriamente, non + solo per il Teorema di Cauchy, ma anche perché solo + l'identità ammette ordine $1$ e perché si è supposto + che nessun elemento abbia ordine $p^2$ (altrimenti + il gruppo sarebbe ciclico). + } $p$ e sia\footnote{ + Tale $h$ deve esistere, altrimenti $G$ sarebbe ciclico. + } $h \in G$ tale che $h \notin \gen{g}$. Per il teorema + precedente $G$ è abeliano, e quindi $\gen{g}\gen{h}$ è + un sottogruppo di $G$. \medskip + + + Inoltre $\gen{g} \cap \gen{h}$ è + banale: se non lo fosse avrebbe ordine $p$, e quindi + $\gen{g}$ e $\gen{h}$ coinciderebbero insiemisticamente, + \Lightning. Pertanto $\gen{g}\gen{h} \cong \gen{g} + \times \gen{h} \cong \ZZ_p \times \ZZ_p$. Infine, poiché + $\abs{\gen{g} \gen{h}} = p^2$, vale anche che + $G = \gen{g} \gen{h}$, da cui la tesi. + \end{example} \end{document} \ No newline at end of file