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\title{Math 720\\Fall 2010\\Homework 2}
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\centerline{\it Due Friday September 17, 2010.}

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\noindent 1. (5 pt) Let $H$ and $K$ be subgroups of $G$. Show that $HK=KH$ if and only if $HK$ is a subgroup of $G$.

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\noindent 2. (5 pt) Let $G$ be a finite group and $H$ a normal subgroup of $G$ such that $\vert H\vert=k$ is relatively prime to $[G:H]$. Show that $H$ is the only subgroup of order $k$.

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\noindent 3. (5 pt) Show that if $G$ is finite and $H$ and $K$ are subgroups such that $\text{gcd}([G:H],[G:K])=1$ then $G=HK$.

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\noindent 4. A subgroup $H\leq G$ is said to be {\it characteristic} if $\sigma(H)\subseteq H$ for all $\sigma\in\text{Aut}(G)$.
\begin{itemize}
\item[a)] (5 pt) Show that any characteristic subgroup of $G$ is normal in $G$.
\item[b)] (5 pt) Show that $G^{\prime}$, the commutator subgroup, is characteristic in $G$ (and hence normal).
\item[c)] (5 pt) Show that if $\phi:G\longrightarrow A$ is a homomorphism of groups, with $A$ abelian, the $\text{ker}(\phi)$ contains the commutator subgroup $G^{\prime}$.
\end{itemize}

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\noindent 5. (5 pt) Let $G$ be a finite group of order $n$ and let $p$ be the smallest prime dividing $n$. Show that if there is a subgroup $H\leq G$ such that $[G:H]=p$, then $H$ is normal in $G$.




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