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{\bf \Large 250A Homework 8}\\[7mm]
{\small Due Monday November 29}\\[4mm]
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\noindent
{\bf Question 1} 
Comment on the (historical) significance of the solvability of $S_5$.

\vspace{.4cm}\noindent
{\bf Question 2} 
Compute the derived series for dihedral groups $D_n$ and their quotients.
Are these groups solvable? Of what length?


\vspace{.4cm}\noindent
{\bf Question 3}
As in question 2 but upper triangular invertible $n\times n$ matrices.


\vspace{.4cm}\noindent
{\bf Question 4}
Find an example of both a finite and infinite group
with a quotient group not isomorphic to any subgroup.

\vspace{.4cm}\noindent
{\bf Question 5}
Let $p$ be prime and $F_p$ the field with p elements\footnote{Take
integers mod $p$ with multiplication and addition defined in the usual way.}. 
\begin{itemize}
\item[(i)] Let $N\leq Gl(3,F_p)$ 
be the group of $3\times 3$ $F_p$-valued upper
triangular matrices
with 1's on the diagonal. Show that $G$ is non-abelian of order $p^3$.
\item[(ii)] Compute the order of $Gl(3,F_2)$. Identify the
  Sylow-$p$-subgroups and decide if any of them are normal.
\end{itemize} 
 


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