\documentclass[a4paper,oneside,fleqn,10pt]{article}
\usepackage[nosolutions]{practicas}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[poly,arrow,curve,matrix]{xy}
\usepackage{array}

\overfullrule5pt

\Materia{Álgebra II}
\Cuatrimestre{Segundo Cuatrimestre --- 2017}
\Practica{9}
\Titulo{El teorema de estructura para módulos finitamente generados sobre un dominio de ideales principales}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Abreviaturas
\newcommand\CC{\mathbb{C}}
\newcommand\RR{\mathbb{R}}
\newcommand\QQ{\mathbb{Q}}
\newcommand\ZZ{\mathbb{Z}}
\newcommand\NN{\mathbb{N}}
\newcommand\FF{\mathbb{F}}
\newcommand\HH{\mathbb{H}}
\newcommand\GG{\mathbb{G}}
\newcommand\II{\mathbb{I}}

\newcommand\Grp{\mathrm{Grp}}
\newcommand\op{\mathrm{op}}
\newcommand\id{\mathsf{id}}
\newcommand\Id{\mathsf{Id}}
\newcommand\Z{\mathsf{Z}}
\newcommand\N{\mathsf{N}}
\newcommand\C{\mathsf{C}}
\renewcommand\O{\mathcal{O}}
\renewcommand\S[1]{[\![#1]\!]}
\newcommand\ot{\otimes}
\newcommand\abs[1]{\lvert#1\rvert}
\newcommand\absx[1]{\left\lvert#1\right\rvert}
\newcommand\gen[1]{\left\langle#1\right\rangle}
\newcommand\gens[1]{\langle#1\rangle}
\newcommand\divides{\mid}
\newcommand\notdivides{\nmid}
\renewcommand\emptyset{\varnothing}
\def\dash---{\thinspace---\hskip.16667em\relax}
\newcommand\p{\mathfrak{p}}
\newcommand\q{\mathfrak{q}}
\newcommand\m{\mathfrak{m}}
\renewcommand\a{\mathfrak{a}}
\renewcommand\k{\Bbbk}


\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\AGL}{AGL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Inn}{Inn}
\DeclareMathOperator{\Out}{Out}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\Spec}{Spec\;}
\DeclareMathOperator{\Hom}{\mathsf{Hom}}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\nil}{nil}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\charact}{char}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% proclaiming things
\theoremseparator{.}
\newtheorem*{Definicion}{Definición}
\theoremindent\parindent
\newtheorem*{Teorema}{Teorema}
\newtheorem*{Proposicion}{Proposición}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hyphenation{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{ejercicios}

%%%%%%%%%%%%
\item Sea $p$ un primo positivo. Clasificar todos los grupos abelianos de orden $p^2$, $p^3$, $p^4$ y $p^5$.


%%%%%%%%%%%%
\item Clasificar los grupos abelianos de orden $18$, $45$, $100$ y $180$.

%%%%%%%%%%%%
\item Sea $G$ un grupo abeliano finito y sea $p$ un
primo positivo que divide al orden de $G$. Probar que el n\' umero de elementos
de orden $p$ en $G$ es coprimo con $p$.


%%%%%%%%%%%
\item  Calcular los coeficientes de estructura de los siguientes grupos abelianos:
\begin{ejitems}
	\item $\ZZ_4\oplus\ZZ_6\oplus\ZZ_9$.
	\item $\ZZ_2\oplus\ZZ_2\oplus\ZZ_8\oplus\ZZ_{14}$.
  \item $\ZZ_2 \oplus \ZZ \oplus \ZZ_{49} \oplus \ZZ$.
  \item $\ZZ _{12}\oplus \ZZ_{21} \oplus \ZZ \oplus \ZZ\oplus \ZZ_{20} \oplus \ZZ_{9} \oplus \ZZ_7$.
	\item Un grupo abeliano $G$ de orden $36$ que tiene exactamente $2$ elementos de orden $3$ y que no tiene elementos de orden $4$.
  \item Un grupo abeliano $G$ de orden $225$ que tiene por lo menos $40$ elementos de orden $15$ y tal que todo subgrupo de orden $9$ de $G$ es isomorfo a $\ZZ_3\oplus\ZZ_3.$
	\end{ejitems}


%%%%%%%%%%%
\item  Calcular los coeficientes de estructura de los siguientes cocientes:
\begin{ejitems}
\item $\ZZ^4 / S$ con $S= \{ m \in \ZZ ^4 : m_1+m_2+m_3+m_4 =0, \ m_1+m_2
-2m_3 = 0\}.$
 \item  $\ZZ^3 / S$ con $S= \{ m \in \ZZ ^3 : m_1 \
{\rm es \ par,} \ m_1+5m_2-m_3 =0\}.$
 \item  $\ZZ^3 / S$ con $S= \{ m \in \ZZ ^3 :
m_1=m_2+m_3\ {\rm es \ par,} \ 3 |m_3\}.$
\end{ejitems}


%%%%%%%%%%%%
\item Caracterizar los $\k[X]$-m\'odulos de dimensi\'on $1$, $2$ y $3$ sobre $\k$, para $\k=\RR,\CC$.

%%%%%%%%%%%%
\item Sea $G\subseteq \ZZ^n$ un subgrupo.
\begin{ejitems}
	\item Probar que $[\ZZ^n:G]$ es finito si y solo si $G$ tiene rango $n$.
	\item Si $G$ tiene rango $n$ y $\{g_1,\dots,g_n\}$ es una base de $g$, sea $M\in M_n(\ZZ)$ la matriz que tiene a $g_i$ como $i$-ésima columna. Probar que $[\ZZ^n:G]=\det M$.
\end{ejitems}

%%%%%%%%%%%%
\item Sean $A= \RR[x]/\langle (x^2+1)^2\rangle$ y $J=\langle x^2+1\rangle\unlhd A$. Probar que todo $A$--módulo finitamente generado es isomorfo a $A^m\oplus (A/J)^n$ para un único par de enteros no negativos $(m,n)$.



\end{ejercicios}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
