From 78bbb4e7ceb6127e8447b7a2c97a6afb493e518e Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 6 Oct 2023 16:53:26 +0200 Subject: [PATCH] feat(algebra1): aggiunge dimostrazione alternativa della card. di HK --- .../main.pdf | Bin 209587 -> 212269 bytes .../main.tex | 13 +++++++++++++ 2 files changed, 13 insertions(+) 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 838db45..234dae0 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 @@ -125,6 +125,19 @@ \] da cui la tesi. \end{proof} + + \begin{proof}[Dimostrazione alternativa] + Si osserva che vale la seguente identità: + \[ HK = \bigcup_{h \in H} hK. \] + Poiché gli $hK$ rappresentano delle classi laterali sinistre + di $G$, se $h' \in H$, o $hK = h'K$ o $hK \cap h'K = \emptyset$. Se $hK = h'K$, allora $h h\inv \in K$, e quindi + $h h\inv \in H \cap K$. Vi sono dunque esattamente + $\abs{H \cap K}$ istanze della classe $hK$ nell'unione + considerata all'inizio della dimostrazione. Allora: + \[ \abs{HK} = \frac{\abs{H} \abs{K}}{\abs{H \cap K}}, \] + dove $\abs{K}$ è il numero di elementi di ogni classe + $hK$. + \end{proof} Pertanto, se le scritture sono uniche, $H \cap K$ deve essere per forza banale (infatti deve valere $\abs{H \cap K} = 1$).