From ccc5bccc4efaba81da501cb6d3ffaae1c14765fa Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Tue, 20 Apr 2021 15:26:07 +0200
Subject: Algebraicas Tema 4

---
 ealg/n2.lyx | 304 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 297 insertions(+), 7 deletions(-)

(limited to 'ealg/n2.lyx')

diff --git a/ealg/n2.lyx b/ealg/n2.lyx
index b2edddd..49dbf78 100644
--- a/ealg/n2.lyx
+++ b/ealg/n2.lyx
@@ -3728,8 +3728,210 @@ grupo de Klein
 \end_inset
 
 .
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Necesario para las demostraciones comentadas.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Dado un grupo 
+\begin_inset Formula $G$
+\end_inset
+
+, llamamos 
+\series bold
+exponente
+\series default
+ o 
+\series bold
+periodo
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Exp}(G)$
+\end_inset
+
+, al menor 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall g\in G,g^{n}=1$
+\end_inset
+
+, o a 
+\begin_inset Formula $\infty$
+\end_inset
+
+ si este no existe.
+ [...] Si un grupo es finito tiene periodo finito [...].
+\end_layout
+
+\begin_layout Plain Layout
+[...] Una 
+\series bold
+descomposición primaria
+\series default
+ o 
+\series bold
+indescomponible
+\series default
+ de un grupo abeliano finito 
+\begin_inset Formula $A$
+\end_inset
+
+ es una expresión de la forma
+\begin_inset Formula 
+\begin{align*}
+A= & \langle a_{11}\rangle_{p_{1}^{\alpha_{11}}}\oplus\dots\oplus\langle a_{1m_{1}}\rangle_{p_{1}^{\alpha_{1m_{1}}}}\oplus\\
+ & \dots\oplus\\
+ & \langle a_{k1}\rangle_{p_{k}^{\alpha_{k1}}}\oplus\dots\oplus\langle a_{km_{k}}\rangle_{p_{k}^{\alpha_{km_{k}}}},
+\end{align*}
+
+\end_inset
+
+donde 
+\begin_inset Formula $p_{1}<\dots<p_{k}$
+\end_inset
+
+ son los primos que dividen a 
+\begin_inset Formula $|A|$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha_{i1}\geq\dots\geq\alpha_{im_{i}}\geq1$
+\end_inset
+
+ para cada 
+\begin_inset Formula $i\in\{1,\dots,k\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+Como 
+\series bold
+teorema
+\series default
+, todo grupo abeliano tiene una descomposición primaria [...].
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+ Para todo grupo finito 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Exp}G\mid|G|$
+\end_inset
+
+, y en particular, para 
+\begin_inset Formula $g\in G$
+\end_inset
+
+, 
+\begin_inset Formula $g^{|G|}=1$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, 
+\begin_inset Formula $\text{Exp}G=\text{mcm}_{g\in G}|g|$
+\end_inset
+
+, pero como para todo 
+\begin_inset Formula $g\in G$
+\end_inset
+
+, 
+\begin_inset Formula $\langle g\rangle$
+\end_inset
+
+ es un subgrupo de 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $|g|\mid|G|$
+\end_inset
+
+, luego 
+\begin_inset Formula $|G|$
+\end_inset
+
+ es múltiplo de todo 
+\begin_inset Formula $|g|$
+\end_inset
+
+ y por tanto lo es de 
+\begin_inset Formula $\text{Exp}G$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
  Dado un cuerpo 
-\begin_inset Formula $K$
+\begin_inset Formula $K\neq0$
 \end_inset
 
 , todo subgrupo finito de 
@@ -3746,6 +3948,98 @@ grupo de Klein
 \end_inset
 
  es cíclico.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración: 
+\series default
+Llamando 
+\begin_inset Formula $m:=\text{Exp}K^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ está formado por raíces de 
+\begin_inset Formula $X^{m}-1$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $|K^{*}|\leq m$
+\end_inset
+
+ y, como 
+\begin_inset Formula $m\mid|K^{*}|>0$
+\end_inset
+
+ (
+\begin_inset Formula $1\in K^{*}$
+\end_inset
+
+), 
+\begin_inset Formula $m=|K^{*}|$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ tiene una descomposición primaria de la forma 
+\begin_inset Formula $\langle a_{1}\rangle_{p_{1}^{\alpha_{1}}}\oplus\dots\oplus\langle a_{k}\rangle_{p_{k}^{\alpha_{k}}}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ distinto, pues si su descomposición primaria es 
+\begin_inset Formula $\bigoplus_{i=1}^{k}\bigoplus_{j=1}^{m_{i}}\langle a_{ij}\rangle_{p_{i}^{\alpha_{ij}}}$
+\end_inset
+
+, el orden es 
+\begin_inset Formula $\prod_{i,j}p_{i}^{\alpha_{ij}}$
+\end_inset
+
+ y el exponente es 
+\begin_inset Formula $\prod_{i}p_{i}^{\alpha_{i1}}$
+\end_inset
+
+, dado que 
+\begin_inset Formula $p_{i}^{\alpha_{ij}}\mid p_{i}^{\alpha_{i1}}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $j$
+\end_inset
+
+, y para que estos coincidan debe ser cada 
+\begin_inset Formula $m_{i}=1$
+\end_inset
+
+.
+ Entonces el orden de 
+\begin_inset Formula $a:=a_{1}+\dots+a_{k}$
+\end_inset
+
+ es 
+\begin_inset Formula $m$
+\end_inset
+
+, luego 
+\begin_inset Formula $\langle a\rangle=K^{*}$
+\end_inset
+
+ es cíclico.
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -4244,7 +4538,7 @@ status open
 \begin_inset Formula $\alpha\in M$
 \end_inset
 
- y por tanto 
+ y 
 \begin_inset Formula $\alpha$
 \end_inset
 
@@ -4345,10 +4639,6 @@ Para
 \end_layout
 
 \end_deeper
-\begin_layout Enumerate
-Ser finitamente generada.
-\end_layout
-
 \begin_layout Standard
 Una propiedad relativa a extensiones es 
 \series bold
@@ -4426,7 +4716,7 @@ LM=LK(\alpha_{1},\dots,\alpha_{n})=L(\alpha_{1},\dots,\alpha_{m}),
 
 \end_inset
 
- pues 
+pues 
 \begin_inset Formula $LK(\alpha_{1},\dots,\alpha_{n})$
 \end_inset
 
-- 
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