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\title{Math 420-620\\Fall 2012\\Homework 8}
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\centerline{\it Due Wednesday October 24, 2012.}

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\noindent 1. (5 pt) Let $G$ be a finite $p-$group of order $p^n$. Show that for all $0\leq k\leq n$, there is a subgroup of $G$ of order $p^k$ and each subgroup of order $p^k$ is normal in a subgroup of order $p^{k+1}$ ($k\leq n-1$).

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\noindent 2. (5 pt) Let $p$ and $q$ be distinct primes with $p<q$ and $q\not\equiv 1\text{mod}(p)$. Show that if $\vert G\vert=pq$ then $G\cong\mathbb{Z}/pq\mathbb{Z}$.

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\noindent 3. (5 pt) Let $G$ be a group. We say that $G$ is \textit{simple} if $G$ contains no normal subgroups except for $G$ itself and the identity. Show that there is no simple group of order 80.

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\noindent 4. Suppose that $G$ is a group of order 72; the goal of this problem is to show that $G$ cannot be simple.
\begin{itemize}
\item[a)] (5 pt) Show that $G$ has either 1 or 4 Sylow 3-subgroups. Conclude that is $G$ is simple, then $G$ must have 4 Sylow 3-subgroups.
\item[b)] (5 pt) Show that if $G$ has 4 Sylow 3-subgroups, then the conjugation action of $G$ on the set of Sylow 3-subgroups induces a homomorphism from $G\longrightarrow\text{S}_4$.
\item[c)] (5 pt) Conclude that $G$ must have a nontrivial normal subgroup (hint: the kernal of a homomorphism is always normal). 
\end{itemize}



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