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\title{Math 720\\Fall 2010\\Exam 1}
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\centerline{\it Due Monday October 11, 2010.}

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\noindent 1. (5 pt) Show that every finitely-generated subgroup of $\mathbb{Q}$ is cyclic.

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\noindent 2. (5 pt) Let $F$ be free on the set $X$ and $n\in\mathbb{N}$. Show that the subgroup of $F$ generated by the set $\{g^n\vert g\in F\}$ is normal in $F$.

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\noindent 3. (5 pt) Let $G$ be a group. Show that $\text{Inn}(G)\cong G/Z(G)$.
 

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\noindent 4. Let $G$ be a group with center $Z(G)$, and $p$ a positive prime integer.
\begin{itemize}
\item[a)] (5 pt) Show that if $G/Z(G)$ is cyclic, then $G$ is abelian.
\item[b)] (5 pt) Use this to show that if $\vert G\vert=p^2$, then $G$ is abelian.
\item[c)] (5 pt) Show that if $\vert G\vert=p^3$ then

\[
Z(G)\cong
\begin{cases}
G &\text{ if $G$ is abelian}\\
\mathbb{Z}_p &\text{ if $G$ is not abelian}
\end{cases}
\]
  
\item[d)] (5 pt) Show that if $\vert G\vert=p^3$ and $G$ is not abelian, then $G/Z(G)\cong\mathbb{Z}_p\oplus\mathbb{Z}_p$.
\end{itemize}



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