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\begin{center}
{\large MATH 150A Winter 2020 \ - \ {\bf Problem Set 5}}
% Put your name and section here.
{\large due February 19}
\end{center}
\begin{enumerate}
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\item Let $C_n$ denote the cyclic group of order $n$.
\begin{enumerate}
\item For which pairs of positive integers $n$ and $m$ is $C_n \times C_m$ cyclic?
\item Prove that $\ZZ \times \ZZ$ is not cyclic.
\end{enumerate}
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\item Let $G$ and $H$ be groups and $\varphi:H \to \Aut(G)$ a homomorphism. The {\em semidirect product group}, $G \rtimes_\varphi H$, is defined as the set $G \times H$ with operation
\[ (g_1,h_1)(g_2,h_2) = (g_1\varphi(h_1)(g_2), h_1h_2). \]
\begin{enumerate}
\item Prove that $G \rtimes_\varphi H$ is a group.
\item Prove that $G \times \{1\}$ is a normal subgroup of $G \rtimes_\varphi H$.
\end{enumerate}
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\item Let $D_n$ denote the dihedral group for a regular $n$-gon with $n \geq 3$. Show that $D_n$ has a semidirect product structure, $$D_n \cong C_n \rtimes_\varphi C_2.$$ What is $\varphi:C_2 \to \Aut(C_n)$ in this case?
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\item (7.1.2) Let $H$ be a subgroup of group $G$. Describe the orbits of the $H$-action on $G$ by left multiplication.
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\item $\OO(n)$ denotes the {\em orthogonal group}, the subgroup of $\GL_n(\RR)$ consisting of all real orthogonal $n\times n$ matrices. These are the rotations and reflections of $\RR^n$ that fix the origin. Find the orbits of the $\OO(2)$-action on $\RR^2$. For a point $(x,y) \in \RR^2$ what is its stabilizer?
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\end{enumerate}
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