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\def\title{Homework 8, April 25}
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\bs  
All problems are due Saturday, May 5, at 4:30 PM.

\bs

{\bf Exam Problem 5 hint:}
First compute the center.  Then, for proving simplicity,
if $\sum a_{ij}x^i y^j$ is a nonzero element of an ideal, try to reduce the
number of nonzero coefficients.
\bs 

\def\dim{\operatorname{dim}}
{\bf Exam Problem 7 hint:} 
Consider various sets upon which the group $G$ acts.  
(1)
It acts on the set of
bases for the vector space $F^n$.  It is easy to count the number of bases.
(2)
A {\sl flag} is a chain of subspaces $0 =
V_0 \subseteq V_1 \subseteq \dots \subseteq V_n = F^n$ with $\dim V_i = i$.
The group $G$ acts on the set of flags, and $H$ is intimately connected with
this action.  It is easy to count the number of flags.
\bye


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