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\title{Math 720\\Fall 2010\\Homework 4}
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\centerline{\it Due Monday October 23, 2010.}

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\noindent 1. (5 pt) Let $G$ be a group and $N$ a normal subgroup of $G$. Suppose that $H$ is a subgroup of $G$ such that $NH=G$ and $N\bigcap H=e$. Show that $G$ is isomorphic to the semidirect product of $N$ and $H$,

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\noindent 2. (5 pt) Show that if $\vert G\vert>2$ then $\text{Aut}(G)$ is nontrivial.

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\noindent 3. (5 pt) Let $p$ be a nonzero prime. Show that if $\vert G\vert=pn$ with $p>n$. Show that $G$ has a normal subgroup of order $p$.

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\noindent 4. Let $p$ and $q$ be distinct primes. Show that there are no simple groups of order:
\begin{itemize}
\item[a)] (5 pt) $pq$,
\item[b)] (5 pt) $p^2q$,
\item[c)] (5 pt) $56$,
\item[d)] (5 pt) $2^33^k$, $k\geq 1$,
\item[e)] (5 pt) $80$.
\end{itemize}



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