From ce0dbb9f31f9f149a3439a77e2dd585ebfe44783 Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 9 Oct 2023 13:57:34 +0200 Subject: [PATCH] feat(algebra1): aggiunge dettagli sul gruppo di permutazione --- .../10. Il gruppo delle permutazioni/main.pdf | Bin 148220 -> 197549 bytes .../10. Il gruppo delle permutazioni/main.tex | 22 ++++++++++-------- .../main.pdf | Bin 184025 -> 185870 bytes .../main.tex | 5 +++- .../3. Azione di coniugio e p-gruppi/main.pdf | Bin 197017 -> 197017 bytes .../main.pdf | Bin 241466 -> 241466 bytes 6 files changed, 16 insertions(+), 11 deletions(-) diff --git a/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.pdf b/Secondo anno/Algebra 1/10. 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1/10. Il gruppo delle permutazioni/main.tex b/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex index dd43b6a..0711933 100644 --- a/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex +++ b/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex @@ -88,14 +88,16 @@ Come corollario di questo risultato, se $m_1$ rappresenta il numero di $1$-cicli di $\sigma$, $m_2$ quello dei suoi $2$-cicli, fino a $m_k$, vale il seguente risultato: \[ \abs{\Cl(\sigma)} = \frac{n!}{m_1! \, 1^{m_1} \, m_2! \, 2^{m_2} \cdots m_k! \, k^{m_k}}, \] - e in particolare esistono tante classi di coniugio quante partizioni di $n$. \medskip - - - Si osserva infine che se $\tau_1 \sigma \tau_1\inv = \tau_2 \sigma \tau_2\inv = \rho$, allora: - \[ \tau_1\inv (\tau_2 \sigma \tau_2\inv) \tau_1 = \tau_1\inv \rho \tau_1 = \sigma, \] - per cui $\tau_1\inv \tau_2 \in Z_{S_n}(\sigma)$ dacché $(\tau_1\inv \tau_2) \sigma = \sigma (\tau_1\inv \tau_2)$. Allora $\tau_1 \in \tau_2 Z_{S_n}(\sigma)$. \medskip - - - Infine, se $H \leq S_n$, $H$ è normale in $S_n$ se e solo se per ogni tipo $H$ contiene - tutte le permutazioni di quel tipo o nessuna. + e in particolare esistono tante classi di coniugio quante partizioni di $n$. + Come conseguenza di questo risultato, per il Teorema orbita-stabilizzatore, + vale che: + \[ \abs{Z_{S_n}(\sigma)} = m_1! \, 1^{m_1} \, m_2! \, 2^{m_2} \cdots m_k! \, k^{m_k}, \] + dove si ricorda\footnote{ + Infatti $Z_{S_n}(\sigma)$ è lo stabilizzatore di $\sigma$ nell'azione di coniugio. + } che due permutazioni coniugano $\sigma$ nella stessa permutazione + $\rho$ se queste due permutazioni fanno parte della stessa classe in $G \quot Z_{S_n}(\sigma)$. Infine, + sempre come corollario dello stesso risultato, + se $H \leq S_n$, $H$ è normale in $S_n$ se e solo se per ogni tipo di + permutazione $H$ contiene + tutte le permutazioni di quel tipo o nessuna. \medskip \end{document} \ No newline at end of file diff --git a/Secondo anno/Algebra 1/2. Azione di un gruppo su un insieme/main.pdf b/Secondo anno/Algebra 1/2. 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