Terminados apuntes de Álgebra Conmutativa

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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+orden
+\series default
+ de [un grupo] 
+\begin_inset Formula $G$
+\end_inset
+
+ al cardinal del conjunto.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un anillo, 
+\begin_inset Formula $(A,+)$
+\end_inset
+
+ es su 
+\series bold
+grupo aditivo
+\series default
+, que es abeliano, y 
+\begin_inset Formula $(A^{*},\cdot)$
+\end_inset
+
+ es su 
+\series bold
+grupo de unidades
+\series default
+, que es abeliano cuando el anillo es conmutativo.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+orden
+\series default
+ de 
+\begin_inset Formula $a\in G$
+\end_inset
+
+ al orden de 
+\begin_inset Formula $\langle a\rangle$
+\end_inset
+
+, 
+\begin_inset Formula $|a|\coloneqq|\langle a\rangle|$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $\langle a\rangle_{n}$
+\end_inset
+
+ para referirnos a 
+\begin_inset Formula $\langle a\rangle$
+\end_inset
+
+ indicando que tiene orden 
+\begin_inset Formula $n$
+\end_inset
+
+.
+ El orden de 
+\begin_inset Formula $a$
+\end_inset
+
+ divide al de 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $f:\mathbb{Z}\to G$
+\end_inset
+
+ el homomorfismo dado por 
+\begin_inset Formula $f(n)\coloneqq a^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $\ker f=n\mathbb{Z}$
+\end_inset
+
+ para algún 
+\begin_inset Formula $n\geq0$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $n=0$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectivo y 
+\begin_inset Formula $(\mathbb{Z},+)\cong\langle a\rangle$
+\end_inset
+
+, y en otro caso 
+\begin_inset Formula $\mathbb{Z}_{n}\cong\langle a\rangle$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $n=|a|$
+\end_inset
+
+ y 
+\begin_inset Formula $a^{n}=1\iff|a|\mid n$
+\end_inset
+
+.
+ De aquí, 
+\begin_inset Formula $a^{k}=a^{l}\iff k\equiv l\bmod n$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $|a|$
+\end_inset
+
+ es el menor entero positivo con 
+\begin_inset Formula $a^{n}=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $a$
+\end_inset
+
+ tiene orden finito y 
+\begin_inset Formula $n>0$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+|a^{n}|=\frac{|a|}{\text{mcd}\{|a|,n\}}.
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $G=\langle a\rangle$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ tiene orden infinito, 
+\begin_inset Formula $G\cong(\mathbb{Z},+)\cong C_{\infty}$
+\end_inset
+
+ y los subgrupos de 
+\begin_inset Formula $G$
+\end_inset
+
+ son los 
+\begin_inset Formula $\langle a^{n}\rangle$
+\end_inset
+
+ con 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $|G|=n$
+\end_inset
+
+, 
+\begin_inset Formula $G\cong(\mathbb{Z}_{n},+)\cong C_{n}$
+\end_inset
+
+ y los subgrupos de 
+\begin_inset Formula $G$
+\end_inset
+
+ son exactamente uno de orden 
+\begin_inset Formula $d$
+\end_inset
+
+ por cada 
+\begin_inset Formula $d\mid n$
+\end_inset
+
+, 
+\begin_inset Formula $\langle a^{n/d}\rangle_{d}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Todos los subgrupos y grupos cociente de 
+\begin_inset Formula $G$
+\end_inset
+
+ son cíclicos.
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $p\in\mathbb{N}$
+\end_inset
+
+ es primo, todos los grupos de orden 
+\begin_inset Formula $p$
+\end_inset
+
+ son isomorfos a 
+\begin_inset Formula $(\mathbb{Z}_{p},+)$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $G=\langle g_{1},\dots,g_{n}\rangle$
+\end_inset
+
+ y 
+\begin_inset Formula $N\unlhd G$
+\end_inset
+
+, 
+\begin_inset Formula $G/N=\langle g_{1}N,\dots,g_{n}N\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema chino de los restos para grupos:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son subgrupos cíclicos de órdenes respectivos 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+, 
+\begin_inset Formula $G\times H$
+\end_inset
+
+ es cíclico si y sólo si 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+ son coprimos.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $g,h\in G$
+\end_inset
+
+ tienen órdenes respectivos 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+ coprimos y 
+\begin_inset Formula $gh=hg$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\langle g,h\rangle$
+\end_inset
+
+ es cíclico de orden 
+\begin_inset Formula $nm$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dados un grupo 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, llamamos 
+\series bold
+conjugado
+\series default
+ de 
+\begin_inset Formula $g\in G$
+\end_inset
+
+ por 
+\begin_inset Formula $a$
+\end_inset
+
+ a 
+\begin_inset Formula $g^{a}\coloneqq a^{-1}ga$
+\end_inset
+
+, y conjugado de 
+\begin_inset Formula $X\subseteq G$
+\end_inset
+
+ por 
+\begin_inset Formula $a$
+\end_inset
+
+ a 
+\begin_inset Formula $X^{a}\coloneqq\{x^{a}\}_{x\in X}$
+\end_inset
+
+.
+ Dos elementos 
+\begin_inset Formula $x,y\in G$
+\end_inset
+
+ o conjuntos 
+\begin_inset Formula $x,y\subseteq G$
+\end_inset
+
+ son 
+\series bold
+conjugados
+\series default
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ si existe 
+\begin_inset Formula $a\in G$
+\end_inset
+
+ con 
+\begin_inset Formula $x^{a}=y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, llamamos 
+\series bold
+automorfismo interno
+\series default
+ definido por 
+\begin_inset Formula $a$
+\end_inset
+
+ al automorfismo 
+\begin_inset Formula $\iota_{a}:G\to G$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\iota_{a}(x)\coloneqq x^{a}$
+\end_inset
+
+.
+ Su inverso es 
+\begin_inset Formula $\iota_{a^{-1}}$
+\end_inset
+
+.
+ El conjugado por 
+\begin_inset Formula $a$
+\end_inset
+
+ de un subgrupo de 
+\begin_inset Formula $G$
+\end_inset
+
+ es otro subgrupo de 
+\begin_inset Formula $G$
+\end_inset
+
+ del mismo orden.
+ [...]
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall g,a,b\in G,g^{ab}=(g^{a})^{b}$
+\end_inset
+
+, y [...] la relación de ser conjugados es de equivalencia.
+ Las clases de equivalencia se llaman 
+\series bold
+clases de conjugación
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+, y llamamos 
+\begin_inset Formula $a^{G}\coloneqq[a]=\{a^{g}\}_{g\in G}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $X$
+\end_inset
+
+ un conjunto.
+ Una 
+\series bold
+acción por la izquierda
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ es una función 
+\begin_inset Formula $\cdot:G\times X\to X$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in X,(\forall g,h\in G,(gh)\cdot x=g\cdot(h\cdot x)\land1\cdot x=x)$
+\end_inset
+
+, y una 
+\series bold
+acción por la derecha
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ es una función 
+\begin_inset Formula $\cdot:X\times G\to X$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in X,(\forall g,h\in G,x\cdot(gh)=(x\cdot g)\cdot h\land x\cdot1=x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\cdot:G\times X\to X$
+\end_inset
+
+ es una acción por la izquierda de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, llamamos 
+\series bold
+órbita
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $G\cdot x\coloneqq\{g\cdot x\}_{g\in G}$
+\end_inset
+
+ y 
+\series bold
+estabilizador
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{Estab}_{G}(x)\coloneqq\{g\in G\mid g\cdot x=x\}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\cdot:X\times G\to X$
+\end_inset
+
+ es una acción por la derecha de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, llamamos órbita de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $x\cdot G\coloneqq\{x\cdot g\}_{g\in G}$
+\end_inset
+
+ y estabilizador de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{Estab}_{G}(x)\coloneqq\{g\in G\mid x\cdot g=x\}$
+\end_inset
+
+.
+ Las órbitas forman una partición de 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Llamamos 
+\series bold
+acción por traslación a la izquierda
+\series default
+ a la acción por la izquierda de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $G/H$
+\end_inset
+
+ dada por 
+\begin_inset Formula $g\cdot xH=gxH$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $G\cdot xH=G/H$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+\text{Estab}_{G}(xH)=[...]=H^{x^{-1}}.
+\]
+
+\end_inset
+
+Análogamente llamamos 
+\series bold
+acción por traslación a la derecha
+\series default
+ a la acción por la derecha de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $H\backslash G$
+\end_inset
+
+ dada por 
+\begin_inset Formula $Hx\cdot g=Hxg$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Cuando 
+\begin_inset Formula $H=1$
+\end_inset
+
+, la acción de traslación es de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+, con 
+\begin_inset Formula $G\cdot x=G$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Estab}_{G}(x)=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La 
+\series bold
+acción por conjugación
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ es la acción por la derecha 
+\begin_inset Formula $x\cdot g\coloneqq x^{g}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $x\cdot G=x^{G}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Estab}_{G}(x)=C_{G}(x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $S$
+\end_inset
+
+ es el conjunto de subgrupos de 
+\begin_inset Formula $G$
+\end_inset
+
+, la 
+\series bold
+acción por conjugación de 
+\begin_inset Formula $G$
+\end_inset
+
+ en sus subgrupos
+\series default
+ es la acción por la derecha de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $S$
+\end_inset
+
+ 
+\begin_inset Formula $H\cdot g=H^{g}$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $X$
+\end_inset
+
+ es un conjunto, 
+\begin_inset Formula $\cdot:S_{n}\times X^{n}\to X^{n}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\sigma\cdot(x_{1},\dots,x_{n})\coloneqq(x_{\sigma(1)},\dots,x_{\sigma(n)})$
+\end_inset
+
+ es una acción por la izquierda.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $\cdot:G\times X\to X$
+\end_inset
+
+ una acción por la izquierda, 
+\begin_inset Formula $H\leq G$
+\end_inset
+
+ e 
+\begin_inset Formula $Y\subseteq X$
+\end_inset
+
+, si 
+\begin_inset Formula $\forall h\in H,y\in Y,h\cdot y\in Y$
+\end_inset
+
+, 
+\begin_inset Formula $\cdot|_{H\times Y}$
+\end_inset
+
+ es una acción por la izquierda de 
+\begin_inset Formula $H$
+\end_inset
+
+ en 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ un grupo actuando sobre un conjunto 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ y 
+\begin_inset Formula $g\in G$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{Estab}_{G}(x)\leq G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $[G:\text{Estab}_{G}(x)]=|G\cdot x|$
+\end_inset
+
+.
+ En particular, si 
+\begin_inset Formula $G$
+\end_inset
+
+ es finito, 
+\begin_inset Formula $|G\cdot x|\mid|G|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si la acción es por la izquierda, 
+\begin_inset Formula $\text{Estab}_{G}(g\cdot x)=\text{Estab}_{G}(x)^{g^{-1}}$
+\end_inset
+
+, y si es por la derecha, 
+\begin_inset Formula $\text{Estab}_{G}(x\cdot g)=\text{Estab}_{G}(x)^{g}$
+\end_inset
+
+.
+ En particular, si 
+\begin_inset Formula $x,g\in G$
+\end_inset
+
+ y 
+\begin_inset Formula $H\leq G$
+\end_inset
+
+, 
+\begin_inset Formula $C_{G}(x^{g})=C_{G}(x)^{g}$
+\end_inset
+
+ y 
+\begin_inset Formula $N_{G}(H^{g})=N_{G}(H)^{g}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $R$
+\end_inset
+
+ es un conjunto irredundante de representantes de las órbitas, 
+\begin_inset Formula $|X|=\sum_{r\in R}|G\cdot r|=\sum_{r\in R}[G:\text{Estab}_{G}(r)]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $G$
+\end_inset
+
+ es un grupo y 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, 
+\begin_inset Formula $|a^{G}|=[G:C_{G}(a)]$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $a^{G}$
+\end_inset
+
+ es unipuntual si y sólo si 
+\begin_inset Formula $a\in Z(G)$
+\end_inset
+
+.
+ 
+\series bold
+Ecuación de clases:
+\series default
+ Si 
+\begin_inset Formula $G$
+\end_inset
+
+ es finito y 
+\begin_inset Formula $X\subseteq G$
+\end_inset
+
+ contiene exactamente un elemento de cada clase de conjugación con al menos
+ dos elementos, entonces 
+\begin_inset Formula $|G|=|Z(G)|+\sum_{x\in X}[G:C_{G}(x)]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado un número primo 
+\begin_inset Formula $p$
+\end_inset
+
+, un 
+\series bold
+
+\begin_inset Formula $p$
+\end_inset
+
+-grupo
+\series default
+ es un grupo en que todo elemento tiene orden potencia de 
+\begin_inset Formula $p$
+\end_inset
+
+, y un grupo finito es un 
+\begin_inset Formula $p$
+\end_inset
+
+-grupo si y sólo si su orden es potencia de 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Cauchy:
+\series default
+ Si 
+\begin_inset Formula $G$
+\end_inset
+
+ es un grupo finito con orden múltiplo de un primo 
+\begin_inset Formula $p$
+\end_inset
+
+, 
+\begin_inset Formula $G$
+\end_inset
+
+ tiene un elemento de orden 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dados un grupo finito 
+\begin_inset Formula $G$
+\end_inset
+
+ y un número primo 
+\begin_inset Formula $p$
+\end_inset
+
+, 
+\begin_inset Formula $H\leq G$
+\end_inset
+
+ es un 
+\series bold
+
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de Sylow
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ si es un 
+\begin_inset Formula $p$
+\end_inset
+
+-grupo y 
+\begin_inset Formula $[G:H]$
+\end_inset
+
+ es coprimo con 
+\begin_inset Formula $p$
+\end_inset
+
+, si y sólo si es un 
+\begin_inset Formula $p$
+\end_inset
+
+-grupo y 
+\begin_inset Formula $|H|$
+\end_inset
+
+ es la mayor potencia de 
+\begin_inset Formula $p$
+\end_inset
+
+ que divide a 
+\begin_inset Formula $|G|$
+\end_inset
+
+.
+ Llamamos 
+\begin_inset Formula $s_{p}(G)$
+\end_inset
+
+ al número de 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupos de Sylow de 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teoremas de Sylow:
+\series default
+ Sean 
+\begin_inset Formula $p$
+\end_inset
+
+ un número primo y 
+\begin_inset Formula $G$
+\end_inset
+
+ un grupo finito de orden 
+\begin_inset Formula $n\coloneqq p^{k}m$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $k,m\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $p\nmid m$
+\end_inset
+
+.
+ Entonces:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $G$
+\end_inset
+
+ tiene al menos un 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de Sylow, que tendrá orden 
+\begin_inset Formula $p^{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $P$
+\end_inset
+
+ es un 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de Sylow de 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ es un 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de 
+\begin_inset Formula $G$
+\end_inset
+
+, existe 
+\begin_inset Formula $g\in G$
+\end_inset
+
+ tal que 
+\begin_inset Formula $Q\subseteq P^{g}$
+\end_inset
+
+.
+ En particular, todos los 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupos de Sylow de 
+\begin_inset Formula $G$
+\end_inset
+
+ son conjugados en 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $s_{p}(G)\mid m$
+\end_inset
+
+ y 
+\begin_inset Formula $s_{p}(G)\equiv1\bmod p$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document