diff options
Diffstat (limited to 'ga/n4.lyx')
| -rw-r--r-- | ga/n4.lyx | 279 |
1 files changed, 197 insertions, 82 deletions
@@ -399,7 +399,8 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open \begin_layout Description \begin_inset Formula $1\implies2\implies3,4\implies5]$ @@ -441,6 +442,11 @@ Si es un grupo con el mismo 1. \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Si \begin_inset Formula $S$ @@ -987,7 +993,8 @@ normal \end_inset . -\end_layout +\begin_inset Note Comment +status open \begin_layout Description \begin_inset Formula $1\iff2\iff3\implies4,5]$ @@ -1087,6 +1094,11 @@ normal Por simetría con los dos anteriores. \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Si \begin_inset Formula $N\leq G$ @@ -1167,10 +1179,10 @@ Si \end_inset tiene índice 2, es normal. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Como las clases por la izquierda módulo \begin_inset Formula $H$ \end_inset @@ -1194,16 +1206,20 @@ Como las clases por la izquierda módulo . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula ${\cal SL}_{n}(\mathbb{R})\unlhd{\cal GL}_{n}(\mathbb{R})$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Si \begin_inset Formula $a,b\in{\cal GL}_{n}(\mathbb{R})$ \end_inset @@ -1219,7 +1235,11 @@ Si si y sólo si tienen igual determinante. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard \series bold @@ -1396,42 +1416,50 @@ Propiedades: Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $f(1)f(1)=f(1\cdot1)=f(1)=f(1)1\implies f(1)=1$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(a)^{-1}=f(a^{-1})$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $f(a)f(a^{-1})=f(aa^{-1})=f(1)=1$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(a_{1}\cdots a_{n})=f(a_{1})\cdots f(a_{n})$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $n=0$ \end_inset @@ -1452,16 +1480,20 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(a^{m})=f(a)^{m}$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $m=0$ \end_inset @@ -1491,7 +1523,11 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $f$ @@ -1502,10 +1538,10 @@ Si \end_inset también. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Sean \begin_inset Formula $x,y\in H$ \end_inset @@ -1525,16 +1561,20 @@ Sean . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $g\circ f:G\to K$ \end_inset es un homomorfismo de grupos. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $a,b\in G$ \end_inset @@ -1546,7 +1586,11 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f^{-1}(H')\leq G$ \end_inset @@ -1566,10 +1610,10 @@ Para \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Claramente \begin_inset Formula $1\in f^{-1}(H')$ \end_inset @@ -1617,13 +1661,13 @@ ab^{-1}=f^{-1}(f(ab^{-1}))=f^{-1}(f(a)f(b)^{-1}), \end_inset . -\begin_inset Newpage pagebreak +\end_layout + \end_inset \end_layout -\end_deeper \begin_layout Enumerate \begin_inset Formula $f$ \end_inset @@ -1633,9 +1677,9 @@ ab^{-1}=f^{-1}(f(ab^{-1}))=f^{-1}(f(a)f(b)^{-1}), \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -1672,6 +1716,7 @@ Como . \end_layout +\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -1709,6 +1754,11 @@ Si \end_layout \end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(G')\leq H$ \end_inset @@ -1732,10 +1782,10 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Claramente \begin_inset Formula $1\in f(G')$ \end_inset @@ -1797,7 +1847,11 @@ Claramente . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard Algunos homomorfismos: \end_layout @@ -2030,13 +2084,6 @@ Si \end_layout \begin_layout Standard -\begin_inset Newpage newpage -\end_inset - - -\end_layout - -\begin_layout Standard Así: \end_layout @@ -2069,10 +2116,10 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout La norma \begin_inset Formula $|\cdot|:\mathbb{C}^{*}\to\mathbb{R}^{*}$ \end_inset @@ -2088,16 +2135,20 @@ La norma , y aplicamos el primer teorema de isomorfía. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula ${\cal GL}_{n}(\mathbb{R})/{\cal SL}_{n}(\mathbb{R})\cong\mathbb{R}^{*}$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout El determinante \begin_inset Formula $\det:{\cal GL}_{n}(\mathbb{R})\to\mathbb{R}$ \end_inset @@ -2113,7 +2164,11 @@ El determinante , y aplicamos el primer teorema de isomorfía. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard En general, \begin_inset Formula $H,K\leq G$ @@ -2124,7 +2179,11 @@ En general, \end_inset . - En efecto, si +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +En efecto, si \begin_inset Formula $\sigma,\tau\in S_{3}$ \end_inset @@ -2171,6 +2230,11 @@ En general, , luego esto no es un grupo. \end_layout +\end_inset + + +\end_layout + \begin_layout Section Orden de un elemento \end_layout @@ -2291,6 +2355,11 @@ Si \end_inset + +\begin_inset Note Comment +status open + +\begin_layout Plain Layout En efecto, sean \begin_inset Formula $m:=|a|$ \end_inset @@ -2314,6 +2383,11 @@ En efecto, sean . \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Si \begin_inset Formula $G=\langle a\rangle$ @@ -3187,10 +3261,10 @@ Sean \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Como \begin_inset Formula $1\cdot x=x$ \end_inset @@ -3219,7 +3293,11 @@ Como . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $[G:\text{Estab}_{G}(x)]=|G\cdot x|$ \end_inset @@ -3234,10 +3312,10 @@ Como \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Sea \begin_inset Formula $H:=\text{Estab}_{G}(x)$ \end_inset @@ -3278,7 +3356,11 @@ Sea . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si la acción es por la izquierda, \begin_inset Formula $\text{Estab}_{G}(g\cdot x)=\text{Estab}_{G}(x)^{g^{-1}}$ @@ -3306,10 +3388,10 @@ Si la acción es por la izquierda, \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Si la acción es por la izquierda, \begin_inset Formula $\text{Estab}_{G}(x)^{g^{-1}}=\{ghg^{-1}:h\cdot x=x\}=\{p\in G:g^{-1}pg\cdot x=x\}=\{p\in G:p\cdot(g\cdot x)=g\cdot x\}=\text{Estab}_{G}(g\cdot x)$ \end_inset @@ -3322,7 +3404,11 @@ Si la acción es por la izquierda, . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $R$ @@ -3333,10 +3419,10 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Se debe a que las órbitas forman una partición de \begin_inset Formula $X$ \end_inset @@ -3344,7 +3430,11 @@ Se debe a que las órbitas forman una partición de . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard Así, si \begin_inset Formula $G$ @@ -3404,7 +3494,17 @@ Dado un número primo \begin_inset Formula $p$ \end_inset -, y por el teorema de Lagrange, un grupo finito es un +, y +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +por el teorema de Lagrange, +\end_layout + +\end_inset + +un grupo finito es un \begin_inset Formula $p$ \end_inset @@ -3430,7 +3530,11 @@ Si \end_inset . - En efecto, si +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +En efecto, si \begin_inset Formula $X\subseteq G$ \end_inset @@ -3466,6 +3570,11 @@ Si . \end_layout +\end_inset + + +\end_layout + \begin_layout Standard \series bold @@ -3488,9 +3597,10 @@ Teorema de Cauchy: \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_layout Standard +\begin_layout Plain Layout \series bold Demostración: @@ -3584,7 +3694,7 @@ Demostración: . \end_layout -\begin_layout Standard +\begin_layout Plain Layout Como \begin_inset Formula $|\sigma|=p$ \end_inset @@ -3670,6 +3780,11 @@ Como . \end_layout +\end_inset + + +\end_layout + \begin_layout Section Teoremas de Sylow \end_layout |