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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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