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{\large MATH 150A Winter 2020 \ - \ {\bf Problem Set 2}}
% Put your name and section here.
{\large due January 24}
\end{center}
\begin{enumerate}
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\item
\begin{enumerate}
\item Characterize the elements of $\CC^\times$ that have order $n$ for positive integer $n$.
\item Characterize the elements of $\CC^\times$ that have order $\infty$.
\end{enumerate}
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\item (2.4.3) Let $a$ and $b$ be elements of a group $G$. Prove that $ab$ and $ba$ have the same order.
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\item (2.4.10) Show by example that the product of elements of finite order in a group need not have finite order. What if the group is abelian?
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\item (2.5.2) Let $H$ and $K$ be subgroups of group $G$.
\begin{enumerate}
\item Prove that the intersection $K \cap H$ is a subgroup of $H$.
\item Prove that if $K$ is a normal subgroup of $G$, then $K \cap H$ is a normal subgroup of $H$.
\end{enumerate}
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\item
\begin{enumerate}
\item Find an injective homomorphism from the symmetric group $S_3$ to $\GL_3(\RR)$.
\item Let $C_8$ denote the cyclic group of order 8. Find an injective homomorphism from $C_8$ to $\GL_2(\RR)$.
\end{enumerate}
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\item (2.5.4) Let $f:\RR^+ \to \CC^\times$ be the map defined by $f(x) = e^{ix}$. Prove that $f$ is a homomorphism, and determine its kernel and image.
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\item Let $D_5$ denote the {\em dihedral group of the pentagon}, which is the group of order 10 consisting of the symmetries of a regular pentagon in the plane. $D_5$ is generated by $r$ and $s$ which represent a counter-clockwise rotation of the pentagon by $2\pi/5$ radians, and a reflection, respectively. Find all subgroups of $D_5$ and determine which subgroups are normal.
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\item
\begin{enumerate}
\item Prove that if $f:\QQ^+ \to \QQ^+$ is a group homomorphism, then $f(x) = cx$ for some constant $c$.
\item Let $V$ and $W$ be vector spaces over $\QQ$ and $T:V\to W$ a function. Prove that $T$ is a group homomorphism between $(V,+)$ and $(W,+)$ if and only if $T$ is a linear map.
\item Is the property in part (a) true for $f:\CC^+ \to \CC^+$?
\item Is the property in part (a) true for $f:\RR^+ \to \RR^+$?
\end{enumerate}
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\end{enumerate}
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