Problem Set 4

(10 points)

(From Gallian)

pp. 35-38: # 12, 16, 17.

Read Chapter 2 of Gallian.

pp. 52: # 2, 3, 5*,

*Read examples 17 and 18 carefully to do this problem. Keep in mind that the multiplication in Gl(2,F) is not necessarily going to use the group structure of Z_11 (it does not use addition mod 11).

Answer in paragraph form:
1. Let f and g be two functions from A to B. What does it mean if these two functions are equal?

2. Suppose that f: A --> B is onto, and g: B--> C is onto. Prove that the composition g f : A-->C onto.

3. What if g:B-->C is onto, but f: A--> B is not onto -- is the composition necessarily onto?

4. Consider the example on page 47 of Gl(2,F) (where F is any of the sets listed above -- F actually stands for field which we will encounter formally toward the end of the course). Multiply two non-trivial elements of the group. Then check that the inverse of a generic element is actually an inverse.Is this group Abelian? Why or why not? Have you encountered it before?