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authorJuan Marín Noguera <juan.marinn@um.es>2020-05-27 12:54:14 +0200
committerJuan Marín Noguera <juan.marinn@um.es>2020-05-27 12:54:14 +0200
commit908911986079fb4bb0414bd035a49c5e6413e3a9 (patch)
treea3ed11e368b6bdbceaa485f8d6c82fd572876931 /ga/n6.lyx
parenta26882b8215c05f19f377009e973cbe013835bd9 (diff)
Comentadas las demostraciones que no entran
Diffstat (limited to 'ga/n6.lyx')
-rw-r--r--ga/n6.lyx166
1 files changed, 122 insertions, 44 deletions
diff --git a/ga/n6.lyx b/ga/n6.lyx
index 4dad294..2ba3ad9 100644
--- a/ga/n6.lyx
+++ b/ga/n6.lyx
@@ -134,6 +134,14 @@ Nos centraremos en los grupos de permutaciones entre conjuntos finitos,
\end_inset
+
+\end_layout
+
+\begin_layout Section
+Ciclos
+\end_layout
+
+\begin_layout Standard
Una
\begin_inset Formula $\sigma\in S_{n}$
\end_inset
@@ -211,6 +219,13 @@ Si
.
\series bold
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
Demostración:
\series default
Para
@@ -290,8 +305,9 @@ Demostración:
.
\end_layout
-\begin_layout Section
-Ciclos
+\end_inset
+
+
\end_layout
\begin_layout Standard
@@ -376,10 +392,10 @@ Dados
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $\sigma(i_{s})=i_{1}$
\end_inset
@@ -403,16 +419,20 @@ Dados
Además se fijan los mismos puntos.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $i_{t}=\sigma^{t-1}(i_{1})$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Para
\begin_inset Formula $t=1$
\end_inset
@@ -436,16 +456,20 @@ Para
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $|\sigma|=s$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Para
\begin_inset Formula $k\in\{1,\dots,s-1\}$
\end_inset
@@ -473,7 +497,11 @@ Para
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Como
\series bold
@@ -935,42 +963,50 @@ El de todos los ciclos.
\begin_layout Enumerate
El de todas las trasposiciones.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $(i_{1}\,\dots\,i_{s})=(i_{1}\,i_{s})(i_{1}\,i_{s-1})\cdots(i_{1}\,i_{2})$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\{(1\,2),(1\,3),\dots,(1\,n)\}$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $(i\,j)=(1\,i)(1\,j)(1\,i)$
\end_inset
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\{(1\,2),(2\,3),\dots,(n-1\,n)\}$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Dados
\begin_inset Formula $j\geq2$
\end_inset
@@ -986,16 +1022,20 @@ Dados
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
\begin_inset Formula $\{(1\,2),(1\,2\,\dots\,n-1\,n)\}$
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Sean
\begin_inset Formula $\tau:=(1\,2)$
\end_inset
@@ -1023,7 +1063,11 @@ Sean
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Si
\begin_inset Formula $p$
@@ -1042,7 +1086,10 @@ Si
\end_inset
.
-
+\end_layout
+
+\begin_layout Standard
+
\series bold
Demostración:
\series default
@@ -1253,6 +1300,11 @@ aplicación signo
, es un homomorfismo de grupos.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
\series bold
Demostración:
\series default
@@ -1275,6 +1327,11 @@ Demostración:
.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Propiedades:
\end_layout
@@ -1284,10 +1341,10 @@ Propiedades:
\end_inset
.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Formula $\text{sgn}(\sigma)\text{sgn}(\sigma^{-1})=\text{sgn}(1)=1$
\end_inset
@@ -1302,13 +1359,17 @@ Propiedades:
.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Toda transposición es impar.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Sean
\begin_inset Formula $\sigma:=(m\,n)$
\end_inset
@@ -1361,7 +1422,11 @@ Sean
inversiones, un número impar.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Enumerate
Si
\begin_inset Formula $\tau_{1},\dots,\tau_{r}$
@@ -1396,10 +1461,10 @@ Un ciclo de longitud
\begin_layout Enumerate
La paridad de una permutación coincide con la del número de componentes
pares de su tipo.
-\end_layout
+\begin_inset Note Comment
+status open
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
Si
\begin_inset Formula $[s_{1},\dots,s_{k}]$
\end_inset
@@ -1425,7 +1490,11 @@ Si
resultado.
\end_layout
-\end_deeper
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Llamamos
\series bold
@@ -1469,7 +1538,11 @@ grupo alternado
\end_inset
y .
- En efecto, es normal por ser el núcleo de un homomorfismo, y el resto es
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, es normal por ser el núcleo de un homomorfismo, y el resto es
consecuencia de que el signo es suprayectivo para
\begin_inset Formula $n\geq2$
\end_inset
@@ -1477,6 +1550,11 @@ grupo alternado
y del primer teorema de isomorfía.
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
Son generadores de
\begin_inset Formula $A_{n}$