From 1900b3a9025e7af2392e5287d4332ac6bf1d5b96 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 17 Oct 2023 15:15:50 +0200 Subject: [PATCH] feat(algebra1): aggiunge risultati su Aut(Z/nZ) e Aut(Z/pZ^n) --- .../1. Il gruppo degli automorfismi/main.pdf | Bin 215463 -> 243582 bytes .../1. Il gruppo degli automorfismi/main.tex | 119 +++++++++++++++++- .../9. I teoremi di isomorfismo/main.pdf | Bin 177183 -> 177196 bytes .../9. I teoremi di isomorfismo/main.tex | 4 +- tex/latex/style/notes_2023.sty | 4 + 5 files changed, 123 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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Il gruppo degli automorfismi/main.tex b/Secondo anno/Algebra 1/1. Il gruppo degli automorfismi/main.tex index 32be564..945391e 100644 --- a/Secondo anno/Algebra 1/1. Il gruppo degli automorfismi/main.tex +++ b/Secondo anno/Algebra 1/1. Il gruppo degli automorfismi/main.tex @@ -148,7 +148,7 @@ bigezioni). \bigskip - \begin{example}[$\Aut(S_3) \cong S_3$] %TODO: aggiungere Aut(Z_2 * Z_2) ~ S_3 + \begin{example}[$\Aut(S_3) \cong S_3$] Si osserva che $Z(S_3)$ deve essere obbligatoriamente banale\footnote{ In generale $Z(S_n)$ è banale per $n \geq 3$. @@ -175,6 +175,39 @@ \end{example} + \begin{example}[$\Aut(\ZZ \quot n \ZZ) \cong (\ZZ \quot n \ZZ)^*$] + Sia $f$ un automorfismo di $\Aut(\ZZ \quot n \ZZ)$. Allora, + necessariamente, $f(\cleq1)$ deve essere un + generatore di $\Aut(\ZZ \quot n \ZZ)$. Si può quindi costruire + un isomorfismo $\zeta : \Aut(\ZZ \quot n \ZZ) \to + (\ZZ \quot n \ZZ)^*$ tale per cui + $f \xmapsto{\zeta} f(\cleq1)$. \medskip + + + Chiaramente $\zeta$ è un omomorfismo, infatti\footnote{ + Potrebbe non risultare completamente ovvio che + valga $f(g(\cleq 1)) = f(\cleq 1) g(\cleq 1)$. + È necessario però ricordarsi che $\ZZ\quot n \ZZ$ è + un gruppo definito sulla somma, e quindi vale sempre + che $f(\cleq a) = a f(\cleq 1) = \cleq{a} f(\cleq 1)$. + }: + \[ \zeta(f \circ g) = f(g(\cleq 1)) = f(\cleq 1) g(\cleq 1) = \zeta(f) \zeta(g). \] + Inoltre $f(\cleq 1) = \cleq 1 \implies f = \Id$, e quindi + $\zeta$ è iniettiva. Infine, per ogni $\cleq a \in (\ZZ \quot n \ZZ)^*$, si può costruire $f_a \in \Aut(\ZZ \quot n \ZZ)$ + di cui è immagine ponendo semplicemente che valga\footnote{ + L'automorfismo è ben determinato dal momento che manda + un generatore in un altro generatore. + } + $f_a(\cleq 1) = \cleq a$. Si conclude quindi che + $\zeta$ è un isomorfismo e dunque che vale il seguente + isomorfismo: + \[ \Aut(\ZZ \quot n \ZZ) \cong (\ZZ \quot n \ZZ)^* \] + Il risultato è valido anche con $n = 0$, da cui si + ricava che: + \[ \Aut(\ZZ) \cong \ZZ^* \cong \{\pm 1\}. \] + \end{example} + + Si illustrano adesso dei risultati molto interessanti sui gruppi di automorfismi dei prodotti diretti, a partire dalla: @@ -237,5 +270,87 @@ per il Teorema cinese del resto e che $\Aut(\ZZ \quot m \ZZ) \cong (\ZZ \quot m \ZZ)^*$, vale che: \[ (\ZZ \quot m \ZZ)^* \times (\ZZ \quot n \ZZ)^* \cong (\ZZ \quot mn \ZZ)^* \] - %TODO: aggiungere dimostrazione Aut(Z/nZ) ~ (Z/nZ)* + + \begin{example}($\Aut((\ZZ \quot p \ZZ)^n)$) + Il gruppo $(\ZZ \quot p \ZZ)^n$ ha una più facile + visualizzazione se lo si pensa come spazio vettoriale su + $\ZZ \quot p \ZZ$ (che per $p$ primo è, per l'appunto, + un campo). In tal caso, gli automorfismi di + $(\ZZ \quot p \ZZ)^n$ coincidono esattamente con gli + endomorfismi invertibili di $\End((\ZZ \quot p \ZZ)^n)$, + e quindi vale in particolare che: + \[ \Aut((\ZZ \quot p \ZZ)^n) \cong \GL_n(\ZZ \quot p \ZZ). \] + In questo modo è estremamente più facile contare il + numero di automorfismi di questo gruppo. È infatti + sufficiente contare le possibili matrici invertibili con + elementi in $\ZZ \quot p \ZZ$. Nella prima colonna di una + matrice $A \in \GL_n(\ZZ \quot p \ZZ)$ possono + essere effettuate $p^n - 1$ scelte (si esclude il vettore + nullo); nella seconda è sufficiente scegliere un vettore + che non stia in $(\ZZ \quot p \ZZ)^n \setminus + \Span(A^1)$, e quindi si hanno $p^n - p$ scelte; per la + terza colonna se ne hanno $p^n - p^2$, ... \medskip + + + Si conclude dunque che vale la seguente identità: + \[ \abs{\Aut((\ZZ \quot p \ZZ)^n)} = \prod_{i=0}^{n-1} (p^n - p^i). \] \medskip + + + Se si prende $m$ \textit{square-free}\footnote{ + Ossia $m$ non è diviso da alcun quadrato; equivalentemente + un primo che compare nella fattorizzazione di $m$ + compare con esponente unitario. + }, il risultato si può estendere facilmente + su $\Aut((\ZZ \quot m)^n)$. Se infatti + $m = p_1 \cdots p_k$, vale che: + \[ + \Aut((\ZZ \quot m \ZZ)^n) \cong + \Aut((\ZZ \quot p_1 \ZZ)^n \times \cdots \times (\ZZ \quot p_k \ZZ)^n) + \cong \Aut((\ZZ \quot p_1 \ZZ)^n) \times + \cdots \times \Aut((\ZZ \quot p_k \ZZ)^n, + \] + dove si è usato sia il Teorema cinese del resto, sia + il fatto per cui $\MCD(p_i, p_j) = 1$ per $i \neq j$. + \end{example} + + \begin{example}[$\Aut(\ZZ \quot 2 \ZZ \times \ZZ \quot 2 \ZZ) \cong S_3$ e altre proprietà] + Ora che è chiara la visualizzazione in senso vettoriale + di $(\ZZ \quot p \ZZ)^n$, si possono elencare alcune + proprietà di $\ZZmod2 \times \ZZmod2$. \medskip + + + Innanzitutto, benché $\ZZmod2 \times \ZZmod2$ sia abeliano, + $\Aut(\ZZmod2 \times \ZZmod2) \cong \GL_2(\ZZmod2)$ non + lo è. Inoltre, ogni sottogruppo proprio e non banale di + $\ZZmod2 \times \ZZmod2$ non è caratteristico: ogni + tale sottogruppo è vettorialmente una retta (infatti + $\ZZmod2 \times \ZZmod2$ ha dimensione due), e quindi + è sufficiente costruire un automorfismo che manda tale + retta in un'altra. \medskip + + + Infine, sempre perché $\Aut(\ZZmod2 \times \ZZmod2) \cong \GL_2(\ZZmod2)$, si può visualizzare facilmente l'isomorfismo + tra $\Aut(\ZZmod2 \times \ZZmod2)$ e $S_3$. Infatti, + $\GL_2(\ZZmod2)$ si compone di $6$ matrici, nella + seguente bigezione con $S_3$: + + \[ + \Matrix{1 & 0 \\ 0 & 1} \bij e, \quad + \Matrix{0 & 1 \\ 1 & 0} \bij (1, 2), \quad + \Matrix{1 & 1 \\ 0 & 1} \bij (2, 3), + \] + + \[ + \Matrix{1 & 0 \\ 1 & 1} \bij (1, 3), \quad + \Matrix{0 & 1 \\ 1 & 1} \bij (1, 2, 3) \quad + \Matrix{1 & 1 \\ 1 & 0} \bij (1, 3, 2). + \] + \vskip 0.1in + + Infine, poiché $\Aut(\ZZmod2 \times \ZZmod2) \cong + S_3 \cong \Aut(S_3)$, $\ZZmod2 \times \ZZmod2$ e + $S_3$ formano un esempio di gruppi non isomorfi + (in particolare uno è abeliano e l'altro no) i cui + gruppi di automorfismo sono isomorfi. + \end{example} \end{document} diff --git a/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.pdf b/Secondo anno/Algebra 1/9. 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