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\begin{center}
{\large MATH 150A Winter 2020 \ - \ {\bf Problem Set 7}}
% Put your name and section here.
{\large due February 28}
\end{center}
\begin{enumerate}
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\item Find a presentation in terms of generators and relations for the following groups.
\begin{enumerate}
\item $\ZZ \times \ZZ$
\item $C_3 \times C_3$
\item $S_3$
\item $A_4$
\end{enumerate}
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\item The group $G = \langle x,y \mid xyx^{-1}y^{-1} \rangle$ is called a {\em free abelian group}. Prove a mapping property of this group: If $u$ and $v$ are elements of an abelian group $A$, there is a unique homomorphism $\varphi:G \to A$ such that $\varphi(x) = u$ and $\varphi(y) = v$.
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\item Let $F$ be the free group on $\{x,y\}$. Prove that the elements $u = x^2$, $v = y^2$, and $z = xy$ generate a subgroup isomorphic to the free group on $\{u,v,z\}$.
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\item Let $F$ be the free group on a nonempty set $S$ with $|S| = k$. How many elements with reduced word of length $n$ does $F$ have?
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\item
\begin{enumerate}
\item Prove that the additive group of $\QQ$ is not finitely generated.
\item Prove that the multiplicative group $\QQ^\times$ is not finitely generated.
\end{enumerate}
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\end{enumerate}
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