From b5d42ec9626f0e34616937fd7ec5583d4959079d Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 23 Oct 2023 10:07:24 +0200 Subject: [PATCH] feat(algebra1): termina la classificazione dei gruppi di ordine pq --- .../12. Il prodotto semidiretto/main.pdf | Bin 199729 -> 238905 bytes .../12. Il prodotto semidiretto/main.tex | 92 +++++++++++++++--- 2 files changed, 80 insertions(+), 12 deletions(-) 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 c1e5898..f8831da 100644 --- a/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex +++ b/Secondo anno/Algebra 1/12. Il prodotto semidiretto/main.tex @@ -78,16 +78,16 @@ $s r^k \xmapsto{\varphi} [h \mapsto sr^k h (sr^k)\inv]$. \end{example} - Si illustra adesso un lemma che verrà riutilizzato successivamente per classificare + Si illustrano adesso due lemmi che verranno riutilizzati successivamente per classificare i gruppi di ordine $pq$. - \begin{nlemma} + \begin{lemma} 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: - \[ \alpha \circ \varphi_k \circ \alpha\inv = \psi_{\beta(k)} \qquad \forall k \in K. \] + \[ \alpha \circ \varphi_k \circ \alpha\inv = \psi_{\beta(k)}, \qquad \forall k \in K. \] Allora vale che $H \rtimes_\varphi K \cong H \rtimes_\psi K$. - \end{nlemma} + \end{lemma} \begin{proof} Si costruisce la mappa $F : H \rtimes_\varphi K \to H \rtimes_\psi K$ tale @@ -99,13 +99,59 @@ desiderato dalla tesi. \end{proof} + \begin{lemma} + Siano $\ZZmod q \rtimes_\varphi \ZZmod p$ e + $\ZZmod q \rtimes_\psi \ZZmod p$ due prodotti + semidiretti con $p$, $q$ primi tali per cui + $p$ è minore di $q$ e $p \mid q-1$. Allora, + se $\varphi$ e $\psi$ sono entrambi omomorfismi non banali, + $\ZZmod q \rtimes_\varphi \ZZmod p$ è isomorfo + a $\ZZmod q \rtimes_\psi \ZZmod p$. + \end{lemma} + + \begin{proof} + Poiché $\ZZmod p$ è ciclico, sia $\varphi$ che + $\psi$ sono univocamente determinati come + omomorfismi da $\varphi_{\cleq 1}$ e + $\psi_{\cleq 1}$. In particolare, affinché i due + omomorfismi non siano banali, gli ordini di + queste valutazioni devono entrambi essere $p$, + dato che $\ord(\varphi_{\cleq 1})$, $\ord(\psi_{\cleq 1}) + \mid \ord(\cleq 1) = p$. \medskip + + + Poiché $\Aut(\ZZmod q) \cong \ZZmod{(q-1)}$ è ciclico, + $\ord(\varphi_{\cleq 1}) = \ord(\psi_{\cleq 1}) \implies \gen{\varphi_{\cleq 1}} = \gen{\psi_{\cleq 1}}$, e quindi + esiste\footnote{ + Si scarta la possibilità in cui $\ell = 0$ dal + momento che altrimenti $\varphi_{\cleq 1}$ sarebbe + l'identità di $\Aut(\ZZmod q)$. + } $\ell \in \{1, \ldots, p-1\}$ tale per cui $\varphi_{\cleq 1} = \psi_{\cleq 1}^\ell$. Si osserva inoltre + che $\psi_{\cleq 1}^\ell = \psi_{\cleq \ell}$. \medskip + + + Sia $\beta \in \Aut(\ZZmod p)$ l'automorfismo\footnote{ + $\beta$ è in effetti un automorfismo dal momento + che $\ell \neq 0$, e quindi $\cleq \ell$ è un altro + generatore di $\ZZmod p$. + } di $\ZZmod p$ univocamente + determinato da $\beta(\cleq 1) = \cleq \ell$. Allora + vale che: + \[ \varphi_{\cleq n} = \varphi_{\cleq 1}^n = + \psi_{\cleq \ell}^n = \psi_{n \cleq \ell} = \psi_{\beta(\cleq n)}, \qquad \forall \cleq n \in \ZZmod p. \] + Si conclude allora per il \textit{Lemma 1} che + $\ZZmod q \rtimes_\varphi \ZZmod p$ è isomorfo + a $\ZZmod q \rtimes_\psi \ZZmod p$. + \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). + di ordine $p$ di $\Aut(\ZZmod q)$ (ossia $\varphi$ non è banale). In particolare esiste un solo gruppo non abeliano + di ordine $pq$ a meno di isomorfismo. \end{proposition} \begin{proof} @@ -119,7 +165,7 @@ 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 + Si osserva che $H \cong \ZZmod q$, $\Aut(H) \cong \ZZmod{(q-1)}$ e analogamente che $K \cong \ZZmod p$. \medskip @@ -130,12 +176,34 @@ 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 + al caso di prima, in cui $G \cong \ZZmod{pq}$), oppure $\abs{\Im \varphi} = p$, e in tal caso $G$ è isomorfo, + per\footnote{ + Infatti $H \cong \ZZmod q$ e $K \cong \ZZmod p$, e + quindi i prodotti semidiretti tra $H$ e $K$ + sono gli stessi di $\ZZmod q$ e $\ZZmod p$. + } il \textit{Lemma 2}, a tutti i prodotti semidiretti + non banali (e quindi, a meno di isomorfismo, ne esiste + soltanto uno). Tale prodotto semidiretto dà luogo ad + un gruppo non abeliano\footnote{ + Se $H \rtimes_\varphi K$ con + $\varphi$ non banale fosse un gruppo abeliano, allora + $\{e\} \times K$ sarebbe normale. Pertanto, + $(h',k')(e,k)(h',k')\inv$ dovrebbe appartenere + a $\{e\} \times K$. Tuttavia vale che: + \[ (h',k')(e,k)(h',k')\inv = + (h', k' k)(\varphi_{{k'}\inv}({h'}\inv), {k'}\inv) = + (h' \varphi_{k}({h'}\inv), k), \] + e quindi dovrebbe valere $\varphi_k(h') = h'$ + per ogni $h' \in H$. In tal caso però + $\varphi_k$ sarebbe l'identità per ogni $k \in K$, + e $\varphi$ sarebbe quindi in particolare banale. + }, e pertanto non può essere + isomorfo a $\ZZmod{pq}$. \end{proof} + In particolare, si osserva che se $G$ non abeliano ha ordine $pq$, + allora $Z(G)$ è banale. Infatti $\abs{Z(G)} \neq p$, $q$ + (altrimenti $G \quot{Z(G)}$ sarebbe ciclico, e quindi + $G$ sarebbe abeliano), né tantomeno $\abs{Z(G)} = pq$. + \end{document} \ No newline at end of file