From 5898380494652fc78d290790114970ad67f7abf0 Mon Sep 17 00:00:00 2001 From: Hearot Date: Tue, 17 Oct 2023 15:33:49 +0200 Subject: [PATCH] feat(algebra1): aggiunge risultato sull'ordine di xy con xy=yx e (ord(x), ord(y))=1 --- .../main.pdf | Bin 212269 -> 220528 bytes .../main.tex | 46 ++++++++++++++++-- 2 files changed, 42 insertions(+), 4 deletions(-) diff --git a/Secondo anno/Algebra 1/6. 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