From 2cdab21aa3da4e5140d6b0a5631c740a5f80cd86 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 22 Oct 2023 16:51:14 +0200 Subject: [PATCH] feat(algebra1): aggiunge delle note sul gruppo dei quaternioni --- .../13. Il gruppo dei quaternioni/main.pdf | Bin 0 -> 87311 bytes .../13. Il gruppo dei quaternioni/main.tex | 54 ++++++++++++++++++ 2 files changed, 54 insertions(+) create mode 100644 Secondo anno/Algebra 1/13. Il gruppo dei quaternioni/main.pdf create mode 100644 Secondo anno/Algebra 1/13. Il gruppo dei quaternioni/main.tex diff --git a/Secondo anno/Algebra 1/13. Il gruppo dei quaternioni/main.pdf b/Secondo anno/Algebra 1/13. 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z7B=zoJ-|%h-`WE+1AZHSVDRtXIT*}z9jtNv`3G-c2-9^)$IoSKpx?(I0Cc@C?N9X} zKoIcPYat*Y+pl8-!UP2WR*#wKnqmLk9t$(`uYC`J^~-PfWnlsTel06(r`vzpgIa4D znd(`?vW=23${IQ9UC&{RG8Pt4l4}HBo85)Y^({!g&-m8|{3I$IyiBmZ1u`=OV83j< y5HLRzm,scale=.7] % rubato a Diego Monaco -- dispense di Algebra 1, p. 52 + \foreach \a/\t in {90/i,-30/j,210/k}{ + \node (\t) at (\a:1cm) {$\t$}; + \draw (\a-20:1cm) arc (\a-20:\a-100:1cm); + } + \end{tikzpicture} + \] + + Moltiplicando in senso orario viene restituito il terzo termine, in senso + antiorario viene restituita la terza potenza del terzo termine rimanente + (per esempio, $ik=j^3=-j$, si ``aggiunge'' in pratica il segno meno). \medskip + + + Si possono classificare molto facilmente i sottogruppi di $Q_8$, che sono: + + \begin{itemize} + \item $Q_8$ stesso, di ordine $8$, banalmente normale, + \item $\gen{i}$, $\gen{j}$ e $\gen{k}$, di ordine $4$, normali perché + di indice $2$, + \item $\gen{-1}$ di ordine $2$, normale perché caratteristico (è l'unico sottogruppo + di ordine $2$ ed è anche il centro $Z(Q_8)$ di $Q_8$), + \item $\{1\}$, di ordine $1$, banalmente normale. + \end{itemize} + + Pertanto $Q_8$ è un esempio di gruppo non abeliano i cui sottogruppi sono tutti normali + (e in particolare anche ciclici). + Inoltre $Q_8$ non può essere decomposto non banalmente in un prodotto semidiretto + tra i suoi sottogruppi: andrebbero infatti scelti due sottogruppi di ordine $4$, che, + essendo normali, indurrebbero obbligatoriamente un prodotto diretto tra gruppi ciclici; + poiché questo prodotto è abeliano, $Q_8$ non può essergli isomorfo. +\end{document} \ No newline at end of file