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{\LARGE\bf Tarea 2\hfill Álgebra Moderna I}

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\begin{enumerate}
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\item Si $H,K \unlhd G$, $H\cap K=\{e\}$ y $\langle H, K\rangle=G$,
muestra que $K$ es isomorfo a $G/H$. 
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\item Sea $H\unlhd G$ tal que $[G:H]=p$ ($p$ primo). Muestra que si
$K\leq G$ no está contenido en $H$ entonces $G=HK$ y $[K:K\cap
H]=p$. 
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\item Si $H,K\unlhd G$ con $G=HK$, muestra que $G/(H\cap K)$ es
isomorfo a $(G/H)\times(G/K)$. 
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\item Si 
$\xymatrix{\mathcal{S}:\;\;1\ar[r]&G'\ar[r]^f&G\ar[r]^g&G''\ar[r]&1}$
es una sucesión de morfismos de grupos, decimos que
\emph{$\mathcal{S}$ es exacta} si $f$ es inyectiva, $g$ es
suprayectiva y $\mathrm{ker}(g)=\mathrm{im}(f)$. Muestra lo siguiente: 
\begin{enumerate}
\item $\mathcal{S}$ es exacta si y sólo si existe $H\unlhd G$ y un
diagrama conmutativo: 
$$
\xymatrix{1\ar[r]&G' \ar[r]^f&
                  G  \ar[r]^g&
                  G''\ar[r]  &1\\
          1\ar[r]&H  \ar@{^(->}[r]\ar[u]|{\varphi}&
                  G  \ar[r]_-{\pi}\ar[u]|{\mathrm{id}}&
                  G/H\ar[r]\ar[u]|{\psi}&1}
$$ 
donde $\varphi$ y $\psi$ son isomorfismos.
\item Si $\mathcal{S}$ es una sucesión exacta con $G'$ y $G''$ grupos
finitamente generados entonces $G$ es finitamente generado. 
\end{enumerate}
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\item Si $G$ es un grupo y 
$$
\mathcal{C}:\;G_n\leq G_{n-1} \leq \dots \leq G_1\leq G_0=G
$$
es una cadena de subgrupos de $G$, decimos que \emph{$\mathcal{C}$ es
una cadena normal de $G$}, si $G_{i+1}\unlhd G_i$ para toda $i$. Si
$\mathcal{C}$ es una cadena normal de $G$, decimos que
\emph{$\mathcal{C}$ es abeliana} (resp. \emph{cíclica}) si
$G_i/G_{i+1}$ es un grupo abeliano (resp. cíclico) para toda $i$.  
\begin{enumerate}
\item
Muestra que si $G\stackrel{f}{\to}G'$ es un morfismo de grupos y
$\mathcal{C}$ es una cadena abeliana (resp. cíclica) de $G'$, entonces
$f^{-1}\big(\mathcal{C}\big)$ es una cadena abeliana (resp. cíclica)
de $G$. 
\item
Muestra que si $G$ es un grupo y $H\unlhd G$, entonces \emph{$G$ es
soluble} (es decir, $G$ tiene una cadena abeliana con $G_n=\{e\}$) si
y sólo si $H$ y $G/H$ son solubles.
\end{enumerate}
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\item Si $G$ es un grupo abeliano finito y $p$ es número primo que
divide al orden de $G$, entonces $G$ tiene un elemento de orden $p$. 
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\end{enumerate}
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