From db667eff2268ecf752d4e55b0bff863eff4af8d4 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 17 Oct 2023 14:15:43 +0200 Subject: [PATCH 1/4] fix(algebra1): riordina i teoremi di isomorfismo secondo l'ordine del corso --- .../9. I teoremi di isomorfismo/main.pdf | Bin 177187 -> 177183 bytes .../9. I teoremi di isomorfismo/main.tex | 64 +++++++++--------- 2 files changed, 32 insertions(+), 32 deletions(-) diff --git a/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.pdf b/Secondo anno/Algebra 1/9. 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I teoremi di isomorfismo/main.tex b/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.tex index 8c839f9..8cc1365 100644 --- a/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.tex +++ b/Secondo anno/Algebra 1/9. I teoremi di isomorfismo/main.tex @@ -60,7 +60,38 @@ \[ \Ker f = \{ gN \mid \varphi(g) = e \} = \{ gN \mid g \in \Ker \varphi \} = \Ker \varphi \quot N. \] - \begin{theorem}[Secondo teorema di isomorfismo, o teorema del diamante] + \begin{theorem}[Secondo teorema di isomorfismo] + Siano $H$ e $N$ due sottogruppi normali di $G$ e sia + $N \leq H$. Allora\footnote{ + Ci sono più modi per vedere che $H \quot N$ è + normale in $G \quot N$. Un modo di vederlo si + ottiene dalla dimostrazione stessa del teorema, + dal momento che si ottiene che $H \quot N$ è + il kernel dell'omomorfismo $\varphi$. Altrimenti, + se $hN \in H \quot N$, $gN hN g\inv N = (ghg\inv)N$, + e poiché $H$ è normale in $G$, $ghg\inv \in H$, da + cui $(ghg\inv)N \in H \quot N$. + }: + \[ \frac{G \quot N}{H \quot N} \cong G \quot H. \] + \end{theorem} + + \begin{proof} + Si costruisce l'omomorfismo $\varphi : G \quot N \to G \quot H$ tale per cui $gN \mapsto gH$. Si verifica innanzitutto + che la mappa $\varphi$ è ben definita: + \[ gnH = gH \impliedby N \subseteq H. \] + Inoltre $\varphi$ è effettivamente un omomorfismo dal momento + che: + \[ \varphi(gkN) = gkH = gH \, kH = \varphi(gN) \varphi(kN). \] + Chiaramente $\varphi$ è una mappa surgettiva e quindi + $\Im \varphi = G \quot H$. + Allora, se $g \in \Ker \varphi$, $\varphi(gN) = gH = H$, e quindi $g \in H$. Pertanto $\Ker \varphi = \{ + gN \mid g \in H + \} = H \quot N$. Si conclude allora, per il Primo teorema + di isomorfismo, che: + \[ \frac{G \quot N}{H \quot N} \cong G \quot H. \] + \end{proof} + + \begin{theorem}[Terzo teorema di isomorfismo, o teorema del diamante] Siano $H$, $N \leq G$ con $N \nsgeq G$. Allora\footnote{ Si osserva che effettivamente $H \cap N$ è normale in $H$. Infatti se $g \in H \cap N$, allora, se @@ -101,35 +132,4 @@ la tesi: \[ H \quot (H \cap N) \cong HN \quot N. \] \end{proof} - - \begin{theorem}[Terzo teorema di isomorfismo] - Siano $H$ e $N$ due sottogruppi normali di $G$ e sia - $N \leq H$. Allora\footnote{ - Ci sono più modi per vedere che $H \quot N$ è - normale in $G \quot N$. Un modo di vederlo si - ottiene dalla dimostrazione stessa del teorema, - dal momento che si ottiene che $H \quot N$ è - il kernel dell'omomorfismo $\varphi$. Altrimenti, - se $hN \in H \quot N$, $gN hN g\inv N = (ghg\inv)N$, - e poiché $H$ è normale in $G$, $ghg\inv \in H$, da - cui $(ghg\inv)N \in H \quot N$. - }: - \[ \frac{G \quot N}{H \quot N} \cong G \quot H. \] - \end{theorem} - - \begin{proof} - Si costruisce l'omomorfismo $\varphi : G \quot N \to G \quot H$ tale per cui $gN \mapsto gH$. Si verifica innanzitutto - che la mappa $\varphi$ è ben definita: - \[ gnH = gH \impliedby N \subseteq H. \] - Inoltre $\varphi$ è effettivamente un omomorfismo dal momento - che: - \[ \varphi(gkN) = gkH = gH \, kH = \varphi(gN) \varphi(kN). \] - Chiaramente $\varphi$ è una mappa surgettiva e quindi - $\Im \varphi = G \quot H$. - Allora, se $g \in \Ker \varphi$, $\varphi(gN) = gH = H$, e quindi $g \in H$. Pertanto $\Ker \varphi = \{ - gN \mid g \in H - \} = H \quot N$. Si conclude allora, per il Primo teorema - di isomorfismo, che: - \[ \frac{G \quot N}{H \quot N} \cong G \quot H. \] - \end{proof} \end{document} \ No newline at end of file From 1900b3a9025e7af2392e5287d4332ac6bf1d5b96 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 17 Oct 2023 15:15:50 +0200 Subject: [PATCH 2/4] 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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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 234dae0..ff0412b 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 @@ -154,11 +154,13 @@ \[ (h,k) \xmapsto{\rho} hk. \] Si osserva che ogni elemento $h$ di $H$ commuta con - ogni elemento $k$ di $K$. Sia infatti $g = k\inv hkh\inv$, allora: + ogni elemento $k$ di $K$. Se infatti si considera il + commutatore $g = [h, k]$, vale che: \[ - g = \underbrace{(k\inv hk)}_{\in H} h\inv \in H, \qquad g = k\inv \underbrace{(hkh\inv)}_{\in K} \in K. + g = \underbrace{(hkh\inv)}_{\in K} k \in K, \qquad g = h\inv \underbrace{(kh\inv k\inv)}_{\in H} \in H. \] - Pertanto $g \in H \cap K \implies hk=kh$. Allora $\rho$ + Pertanto $g \in H \cap K \implies [h, k] = e \implies + hk=kh$. Allora $\rho$ è un omomorfismo, infatti: \[ \rho((hh',kk')) = hh'kk' = hkh'k' = \rho((h,k)) \rho((h',k')). @@ -173,7 +175,43 @@ 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 + $G \cong G_1' \times G_2'$. \medskip + + + In particolare vale il seguente risultato, considerando + $\gen{x} \cap \gen{y} = \{e\}$: + + \begin{proposition} + Siano\footnote{ + In generale, se $\MCD(\ord(x), \ord(y)) > 1$, + non vale che $\ord(xy) = \mcm(\ord(x), \ord(y))$, + benché sicuramente $\ord(xy) \mid \mcm(\ord(x), \ord(y))$, + sempre a patto che $x$ e $y$ commutino. + È sufficiente considerare in $\ZZmod6$ gli elementi + $\cleq 1$ e $\cleq 2$: infatti $\ord(\cleq 1) = 6$ e + $\ord(\cleq 2) = 3$, ma $\ord(\cleq 1 + \cleq 2) = + \ord(\cleq 3) = 2 \neq 6$. + }\footnote{ + A prescindere da quanto valga $\MCD(\ord(x), \ord(y))$, + se $x$ e $y$ commutano, esiste sempre un elemento + $g \in G$ tale per cui $\ord(g) = \mcm(\ord(x), \ord(y))$. + } $x$, $y$ due elementi di $G$ che commutano con + $\MCD(\ord(x), \ord(y)) = 1$. Allora $\ord(xy) = \ord(x) \ord(y)$. % TODO: aggiungere la dimostrazione + \end{proposition} + + \begin{proof} + Chiaramente $\ord(xy) \mid \ord(x) \ord(y)$, dal momento + che $(xy)^{\ord(x) \ord(y)} = x^{\ord(x) \ord(y)} y^{\ord(x) \ord(y)} = e$, dove si è usato che $x$ e $y$ commutano. + Sia allora $k = \ord(xy)$. Vale allora che + $x^k y^k = e \implies x^k = y^{-k} \in \gen{x} \cap \gen{y}$. + Tuttavia $\abs{\gen{x} \cap \gen{y}} \mid + \MCD(\abs{\gen{x}}, \abs{\gen{y}}) = \MCD(\ord(x), \ord(y)) = + 1$, e quindi $\gen{x} \cap \gen{y} = \{e\}$. Allora + deve valere che $x^k = y^{-k} = e \implies + \ord(x), \ord(y) \mid k$, da cui si deduce che + $\ord(x) \ord(y) \mid k = \ord(x) \ord(y)$. Si conclude dunque che + $\ord(xy) = \ord(x) \ord(y)$. + \end{proof} \vskip 0.2in Si può adesso dimostrare il seguente fondamentale From 09cda00a966fcec4456b0a925264ff8ca95d6939 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 17 Oct 2023 15:44:11 +0200 Subject: [PATCH 4/4] feat(algebra1): aggiunge esempio del teorema di corrispondenza --- .../main.pdf | Bin 171123 -> 175373 bytes .../main.tex | 23 +++++++++++++++++- 2 files changed, 22 insertions(+), 1 deletion(-) diff --git a/Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.pdf b/Secondo anno/Algebra 1/7. 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