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{\LARGE\bf Tarea-Examen 3\hfill Álgebra Moderna I}

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\begin{enumerate}
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\item Sea $G$ un grupo y $H\leq G$. Muestra que $H$ es normal en $G$
si y sólo si $H$ es unión de clases de conjugación de $G$. 
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\item Sea $G$ un grupo y $X$ un $G$-conjunto. Si $x,y\in X$ están en
la misma orbita, da un isomorfismo entre el estabilizador de $x$ y el
estabilizador de $y$.
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\item Muestra que si $G$ es un grupo finito y $H\leq G$ con $[G:H]=p$,
donde $p$ es el primo más pequeño que divide al orden de $G$, entonces
$H$ es normal en $G$.
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\item Muestra que si $G$ es un grupo finito de orden $pn$, donde $p$ es
número primo mayor o igual que $n$, entonces $G$ tiene un subgrupo
normal de orden $p$.  
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\item Si $G$ es grupo no abeliano de orden $p³$, muestra lo siguiente:
\begin{enumerate}
\item $\mathrm{Z}(G)$ es cíclico de orden $p$. 
\item $G/\mathrm{Z}(G)$ es isomorfo a $\mathrm{Z}(G)\times
\mathrm{Z}(G)$.
\item Si $g^p\equiv 1$ para todo $g\in G$, entonces $G$ tiene un
subgrupo de orden $p²$.  
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{\bf SUERTE!!!}
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