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\title{Math 420-620\\Fall 2012\\Homework 2}
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\centerline{\it Due Monday September 10, 2012.}

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\noindent 1. (5 pt) Let $G$ be a finite group. Show that any element of $G$ has finite order.


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\noindent 2. (5 pt) Show that any group of exponent 2 is abelian.

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\noindent 3. The goal of this problem is to show that any {\it finite} group generated by two elements of order two is dihedral (with $\mathbb{Z}_2\times\mathbb{Z}_2$ being considered a ``degenerate" dihedral group). Suppose that $G$ is generated by the elements $x,y\in G$, both of order $2$.
\begin{itemize}
\item[a)] (5 pt) Assuming that the order of $G$ is finite, what can you say about the order of the element $xy\in G$?
\item[b)] (5 pt) Show that the group generated by $x$ and $y$ is the same as the group generated by $xy$ and $y$
\item[c)] (5 pt) Show that the group generated by $x$ and $y$ is dihedral.
\end{itemize}

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\noindent 4. (5 pt) Let $G$ be a group. Recall that the center of $G$ is defined by $\text{Z}(G)=\{z\in G\vert zg=gz,\ \forall g\in G\}$. Compute $\text{Z}(\text{D}_n)$.



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