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\section*{Math 3140: Homework 11}


\begin{enumerate}
\item[17.1.]  Let $G=\langle (1,2,3)(4,5), (7,8)\rangle\subseteq S_8$.  Then $G$ acts on the set $X=\{1,2,3,4,5,6,7,8\}$.  Calculate the orbits of the $G$-action in $X$, and the stabilizers of $1$, $2$ and $7$.
\item[17.4] Let $G$ act on a set $X$, and let $O$ be an orbit in $G$.  Show that $\Stab_G(x)=\Stab_G(y)$ for all $x,y\in O$ if and only if $\Stab_G(x)\triangleleft G$.
\item[17.7-8.]  \textbf{The diagonal action.} Suppose $G$ acts on $X$ and $Y$.   Let
\begin{equation} \begin{array}{ccc} G\times (X\times Y)  & \longrightarrow & X\times Y\\ (g,(x,y)) & \mapsto & (g(x),g(y))\end{array} \tag{$\ast$}\end{equation}
\begin{enumerate}
\item Show that ($\ast$) gives an action of $G$ on $X\times Y$. 
\item For example, find the orbits of $G=\langle (1,2,3,4),(2,4)\rangle\subseteq S_4$ acting on $X\times X$ diagonally, where $X=\{1,2,3,4\}$.  Is this action transitive?
\item In general, find the stabilizer of $(x,y)\in X\times Y$ (in terms of the stablizers of $x$ and $y$).     
\end{enumerate}
\item[17.10.] The \emph{centralizer} of $g\in G$ is the subgroup
$$C_G(g)=\{h\in G\mid hgh^{-1}=g\}.$$
\begin{enumerate}
\item Identify a set $X$ with an action of $G$ on that set such that $C_G(g)=\Stab_G(g)$ for that action.
\item Show that if some conjugacy class of $G$ has exactly two elements, then $G$ is not simple.
\end{enumerate}
\end{enumerate}
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