From d6edf56c92b430f7154218cc8221e9d90358405f Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 20 Jul 2023 20:40:02 +0200 Subject: [PATCH] feat(algebra1): aggiunge la proposizione sulla catena di sottogruppi normali in un p-gruppo --- .../main.pdf | Bin 156852 -> 171123 bytes .../main.tex | 50 ++++++++++++++++++ 2 files changed, 50 insertions(+) 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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anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.tex b/Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.tex index 14e20ec..2a6cde8 100644 --- a/Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.tex +++ b/Secondo anno/Algebra 1/7. Il teorema di corrispondenza e catene di sottogruppi normali/main.tex @@ -92,5 +92,55 @@ $k \in K$ tale per cui $x=yk$. Allora: \[ xH(K \quot H) = yH \, kH (K \quot H) = yH(K \quot H), \] da cui la tesi. + \end{proof} \bigskip + + + Si illustra allora il seguente fondamentale risultato sui $p$-gruppi, + che è conseguenza del Teorema di corrispondenza e delle proprietà degli + ordini di gruppi abeliani. + + \begin{proposition} + Sia $G$ un $p$-gruppo di ordine $p^n$, con $n \in \NN^+$. Allora + esiste una successione $H_1$, ..., $H_{n-1}$ di sottogruppi normali in $G$ + tali per cui: + \[ \{ e \} < H_1 < H_2 < \cdots < H_{n-1} < G, \qquad \abs{H_i} = p^i. \] + \end{proposition} + + \begin{proof} + Si dimostra la tesi per induzione su $n$. Per $n=1$, la tesi è banale. Si + ipotizzi allora che la tesi valga per $t