From f7e38332ca29dad4598a7a01d83b7846481b4149 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 22 Oct 2023 16:17:37 +0200 Subject: [PATCH] feat(algebra1): aggiunge i risultati principali sui prodotti semidiretti --- .../12. Il prodotto semidiretto/main.pdf | Bin 150364 -> 199729 bytes .../12. Il prodotto semidiretto/main.tex | 89 +++++++++++++++++- tex/latex/style/notes_2023.sty | 2 + 3 files changed, 90 insertions(+), 1 deletion(-) diff --git a/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.pdf b/Secondo anno/Algebra 1/12. 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Il prodotto semidiretto/main.tex b/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex index f7c2cd2..c1e5898 100644 --- a/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex +++ b/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex @@ -34,7 +34,14 @@ un gruppo in un prodotto semidiretto di due suoi sottogruppi: \begin{theorem}[di decomposizione in prodotto semidiretto] - Siano $H$ e $K$ due sottogruppi di $G$ con $H \cap K = \{e\}$ e + Siano\footnote{ + Si osserva che questo teorema richiede \textit{quasi} le stesse ipotesi + del Teorema di decomposizione in prodotto diretto. L'unica ipotesi che + manca è quella della normalità di $K$. Ciononostante, questo teorema + copre anche il teorema analogo sul prodotto diretto: se $K$ fosse normale, + $\varphi$ sarebbe l'identità ($h$ e $k$ commuterebbero), e quindi + $H \rtimes_\varphi K$ sarebbe esattamente $H \times K$. + } $H$ e $K$ due sottogruppi di $G$ con $H \cap K = \{e\}$ e $H \nsgeq G$. Allora vale che $HK \cong H \rtimes_\varphi K$ con $\varphi : K \to \Aut(H)$ tale per cui\footnote{ Tale mappa è ben definita dal momento che $H$ è normale in $G$. @@ -51,4 +58,84 @@ Chiaramente $\alpha$ è iniettivo dal momento che $hk=e \implies h = k\inv \in H \cap K \implies h = k = e$. Infine $\alpha$ è surgettiva dal momento che $hk = \alpha(h, k)$, e quindi $\alpha$ è un isomorfismo. \end{proof} + + \begin{example}[$\Sn \cong \An \rtimes_\varphi \gen{\tau}$] + Sia $\tau$ una trasposizione di $\Sn$. Allora $\An$ è normale in $\Sn$, + $\An \cap \gen{\tau} = \{e\}$ e $\abs{\An} \abs{\gen{\tau}} = \abs{\Sn} \implies + \Sn = \An \gen{\tau}$. Allora, per il Teorema di decomposizione in prodotto + semidiretto, vale che: + \[ \Sn \cong \An \rtimes_\varphi \gen{\tau}, \] + con $\varphi : \gen{\tau} \to \Aut(\An)$ tale per cui + $\tau \xmapsto{\varphi} [h \mapsto \tau h \tau\inv]$. + \end{example} + + \begin{example}[$\Dn \cong \rotations \rtimes_\varphi \gen{sr^k}$] + Sia $sr^k$ una qualsiasi simmetria di $\Dn$. Allora $\rotations$ è normale in $\Dn$, + $\rotations \cap \gen{sr^k} = \{e\}$ e $\abs{\rotations} \abs{\gen{s r^k}} = \abs{\Dn} \implies + \Dn = \rotations \gen{s r^k}$. Allora, come prima, vale che: + \[ \Dn \cong \rotations \rtimes_\varphi \gen{sr^k}, \] + con $\varphi : \gen{s r^k} \to \Aut(\rotations)$ tale per cui + $s r^k \xmapsto{\varphi} [h \mapsto sr^k h (sr^k)\inv]$. + \end{example} + + Si illustra adesso un lemma che verrà riutilizzato successivamente per classificare + i gruppi di ordine $pq$. + + \begin{nlemma} + Siano $\varphi$, $\psi : K \to \Aut(H)$ tali per cui esistono + $\alpha \in \Aut(H)$ e $\beta \in \Aut(K)$ che soddisfano la seguente + identità: + \[ \alpha \circ \varphi_k \circ \alpha\inv = \psi_{\beta(k)} \qquad \forall k \in K. \] + Allora vale che $H \rtimes_\varphi K \cong H \rtimes_\psi K$. + \end{nlemma} + + \begin{proof} + Si costruisce la mappa $F : H \rtimes_\varphi K \to H \rtimes_\psi K$ tale + per cui $(h, k) \xmapsto{F} (\alpha(h), \beta(k))$. Si verifica che $F$ è un omomorfismo: + \[ F(h \varphi_k(h'), k k') = (\alpha(h) \alpha(\varphi_k(h')), \beta(k) \beta(k')), \] + e quindi, poiché $\alpha \circ \varphi_k = \psi_{\beta(k)} \circ \alpha$: + \[ F(h \varphi_k(h'), k k') = (\alpha(h)\psi_{\beta(k)}(\alpha(h')), \beta(k) \beta(k')) = F(h, k) F(h', k'). \] + Chiaramente $F$ è anche iniettiva e surgettiva, e quindi $F$ è l'isomorfismo + desiderato dalla tesi. + \end{proof} + + \begin{proposition} + Sia $G$ un gruppo di ordine $pq$ con $p$ e $q$ primi tali per cui $p < q$. Allora $G$ è + isomorfo a $\ZZ_{pq}$ se $p \nmid q-1$. Altrimenti $G$ è isomorfo a $\ZZmod{pq}$ o + a $\ZZmod q \rtimes_\varphi \ZZmod p$ con $\varphi : \ZZmod p \to \Aut(\ZZmod q)$ + univocamente determinata dalla relazione + $\cleq 1 \xmapsto{\varphi} f$ con $f$ un qualsiasi elemento + di ordine $p$ di $\Aut(\ZZmod q)$ (ossia $\varphi$ non è banale). + \end{proposition} + + \begin{proof} + Per il Teorema di Cauchy, esistono due elementi $x$ e $y$ di $G$ con $\ord(x)=q$ + e $\ord(y)=p$. Siano $H = \gen{x}$ e $K = \gen{y}$. Allora, poiché $[G : H] = p$ + è il più piccolo primo che divide $\abs{G} = pq$, $H$ è normale. Inoltre + $H \cap K = \{e\}$, dacché $\abs{H \cap K} \mid \MCD(p, q) = 1$. Pertanto + $\abs{HK} = \abs{H} \abs{K} = pq \implies G = HK$. \medskip + + + Per il Teorema di decomposizione di un gruppo in un prodotto semidiretto, + $G$ è isomorfo al prodotto semidiretto $H \rtimes_\varphi K$ con + $\varphi : K \to \Aut(H)$ tale per cui $k \xmapsto{\varphi} [h \mapsto k h k\inv]$. + Si osserva che $H \cong \ZZmod q$, $\Aut(H) \cong \ZZmod{q-1}$ e analogamente che + $K \cong \ZZmod p$. \medskip + + + Deve inoltre valere anche che $\abs{\Im \varphi} \mid \MCD(\abs{K}, + \abs{\Aut(H)}) = \MCD(p, q-1)$. Pertanto, se $p \nmid q-1$, $\MCD(p, q-1) = 1$, + e quindi $\Im \varphi$ è banale. In tal caso $\varphi$ è la mappa che associa + ogni $k$ all'identità di $\Aut(H)$, e quindi $G \cong H \times K \cong \ZZmod p \times \ZZmod q \cong \ZZmod{pq}$, dove si è usato il Teorema cinese del resto. \medskip + + + Altrimenti $\MCD(p, q-1) = p$, e quindi $\Im \varphi$ può essere banale (riconducendoci + al caso di prima, in cui $G \cong \ZZmod{pq}$), oppure $\abs{\Im \varphi} = p$. Si + mostra adesso che i prodotti semidiretti su $\varphi$ non banale sono tutti isomorfi + a prescindere dalla scelta di $\varphi$. \medskip + + + %TODO: terminare la discussione del caso in cui p divide q-1 + \end{proof} + \end{document} \ No newline at end of file diff --git a/tex/latex/style/notes_2023.sty b/tex/latex/style/notes_2023.sty index b5f6e52..3350567 100644 --- a/tex/latex/style/notes_2023.sty +++ b/tex/latex/style/notes_2023.sty @@ -222,6 +222,7 @@ % Spesso utilizzati durante il corso di Algebra 1 +\newcommand{\Dn}{D_n} \newcommand{\Sn}{S_n} \newcommand{\bij}{\leftrightarrow} @@ -257,6 +258,7 @@ \newtheorem*{example}{Esempio} \newtheorem{exercise}{Esercizio} \newtheorem{lemma}{Lemma} +\newtheorem*{nlemma}{Lemma} \newtheorem*{note}{Nota} \newtheorem*{remark}{Osservazione} \newtheorem*{proposition}{Proposizione}