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\section*{Math 403/503 Spring 2024}
\subsection*{Homework 4, due February 14}
\begin{enumerate}
\item Let $G$ be the additive group of real numbers. Let the action of $\theta\in G$ on the real plane $\mathbb{R}^2$ be given by rotating the plane counterclockwise about the origin through $\theta$ radians. Let $P$ be a point on the plane other than the origin.
\begin{enumerate}[{\rm (a)}]
\item Show that this rule defines an action.
\item Describe geometrically the orbit containing $P$.
\item Find the stablizer subgroup $G_P$ of $P$.
\end{enumerate}
\smallskip
\item Let $G$ be a group acting on a set $X$. Show that $gG_xg^{-1}=G_{g.x}$ for all $g\in G$ and all $x\in X$.
\smallskip
\item Let $G=A_4$ and let $G$ act on itself by conjugation.
\begin{enumerate}[{\rm (a)}]
\item Determine the orbits (conjugacy classes) of each element in $G$.
\item For each $\sigma\in G$, find the stabilizer subgroup $G_\sigma$ (centralizer).
\end{enumerate}
\smallskip
\item Let $Q_8=\{\pm 1, \pm i,\pm j,\pm k\}$ be the quaternion group of order $8$.
\smallskip
\begin{enumerate}[{\rm (a)}]
\item Find $|Z(G)|$.
\item Find a representative $g_i$ for each non-trivial conjugacy class of $Q_8$. 
\item Compute $[G:C_G(g_i)]$ and verify that the class equation holds for $G$.
\end{enumerate}
\smallskip
\item Suppose that the vertices of a regular hexagon are to be colored either cardinal or gold. How many ways can this be done up to a symmetry of the hexagon (from the dihedral group $D_6$)?
\end{enumerate}

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