# Math 817: Problem set 8

*Instructions*: Write up any four of the following problems (but
you should *try* all of them, and
in the end you should be sure you know how to do all of them).
Your write ups are due Friday, October 29, 2004.
Each problem is worth 10 points, 9 points for correctness
and 1 point for communication. (Your goal is not only
to give correct answers but to communicate
your ideas well. Make sure you use good English,
so proofread your solutions.
Once you finish a solution, you
should restructure awkward sentences, and strike out
anything that is not needed in your approach to the problem.)
- Prove that {I, -I} is the unique normal subgroup of GL
_{2}(**R**)
of order 2.
- Let
**R** denote the field of real numbers.
- (a) Prove that GL
_{2}(**R**) is not isomorphic
to SL_{2}(**R**) x (**R**^{x}). You may apply
Problem 1, even if you do not turn that one in.
- (b) Prove that O
_{2} is not isomorphic
to SO_{2} x {I, -I}. [Hint: mimic your proof to part (a).]

- Let A be an element of SO
_{3} with angle *a*.
Prove that cos(*a*) = (tr(A) - 1)/2.
- Do #6.1, on p. 149 of Artin.
- Do #6.5, on p. 149 of Artin. Assume the field
**C** of
complex numbers for part (b); for the other parts, assume the field
**R** of real numbers.
- A square matrix A is
*nilpotent* if A^{n} = 0
for some positive integer n. If A is a nilpotent square matrix,
prove that det(I + A) = 1.