# Takehome Midterm

```Takehome Midterm
Math 332, Spring 2013
These problems must be written up in LATEX, and are due this Sunday, April 14.
Rules: This is a midterm exam, not a homework assignment. You must solve the problems
entirely on your own, and you should not discuss the problems with any other students in
the class, or with anyone on the internet. When working on the problems, you may use the
textbook, your class notes, the materials on the class web page, and any textbook on Linear
Algebra, but you may not consult any other sources, including internet websites.
1. Let G be a group, and suppose that aba3 b2 a = e for all a, b ∈ G. Prove that G is
trivial.
2. (a) List the possible cycle structures for an element of A7 .
(b) Make a table showing the number of elements of A7 with each cycle structure.
3. Let n ∈ N, let v ∈ Rn be a nonzero vector, and let
G = {A ∈ GL(n, R) | v is an eigenvector for A}.
Prove that G is a subgroup of GL(n, R).
4. List the eight automorphisms of D4 . Express your answers as permutations of the set
{e, r, r2 , r3 , s, rs, r2 s, r3 s}.
5. Let G =
a a − 1 a ∈ (0, ∞) .
0
1
(a) Prove that G is a subgroup of GL(2, R).
(b) Prove that G is isomorphic to R (the group of real numbers under addition).
```