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{\bf \Large 250A Homework 5}\\[7mm]
{\small Due Monday November 8}\\[4mm]
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\noindent
{\bf Question 1} 
Subgroups $H\leq S_4$ act on $S\equiv \{1,2,3,4\}$ in the natural way.
For each subgroup $H$ below, compute the orbits and stabilizer
of each $s\in S$:
\begin{enumerate}
\item[(i)] $H=\langle(123)\rangle$
\item[(ii)] $H=\langle(1234)\rangle$
\item[(iii)] $H=\langle(12),(34)\rangle$
\item[(iv)] $H=D_8$
\item[(v)] $H=A_4$
\end{enumerate}

\vspace{.4cm}\noindent
{\bf Question 2} 
Let $G={\mathbb R}\ni r$ be the group of real numbers under addition.
Show that 
\begin{enumerate}
\item[(i)] $S={\mathbb R}^n\ni x\mapsto x+r v $ ($v\in {\mathbb R}^2$
           fixed),
\item[(ii)] $S={\mathbb C}\ni \rho e^{i\theta}\mapsto
\rho e^{i(\theta+r)}$
\end{enumerate}  
are group actions and describe their orbits geometrically. 
What do stabilizers look like?

\vspace{.4cm}\noindent
{\bf Question 3} 
Let $H\leq G$ with $[G:H]=n$. Show that $G$ has a normal subgroup of
index at most $n!$.

\vspace{.4cm}\noindent
{\bf Question 4} Find the center of $Gl(2,{\mathbb R})$, the dihedral groups,
and $SO(3,{\mathbb R})$.

\vspace{.4cm}\noindent
{\bf Question 5}
Show that $G/Z(G)$ cyclic $\Rightarrow$ $G$ is abelian. Show that if $p$ is
prime, then all groups of order $p^2$ are abelian.

\vspace{.4cm}\noindent
{\bf Question 6} {\it Research problem -- answers must be less than 1 page}.
What are (i) a Lie group, (ii) a Lie algebra and (iii) a Dynkin diagram?

 
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