diff options
Diffstat (limited to 'ga/n6.lyx')
| -rw-r--r-- | ga/n6.lyx | 166 |
1 files changed, 122 insertions, 44 deletions
@@ -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 $A_{n}$ |