Math 529, Advanced Geometry, Spring 1999
Homework for weeks 3 and 4

Due date	Do	Hand in
Thurs. April 22	Ch. 2, problems 4, 5, 13, 14;	problem D below
	at least one of 1, 2, 3, and 6;	Extension: Ch. 2 problem 7
	at least one of 8, 9, and 10
	Ch. 3 problem 6;	Ch. 3 problems 9 and 12
	Note: first midterm is April 22.
Thurs. April 29	Ch. 3, problems 1-5, 8, 11	Ch. 3 problem 10
		Extension: Ch. 3 problem 7

D.

Let

$$\alpha_1((x,y)) = (x,-y) \ \ \ {\rm and} \ \ \ \alpha_2((x,y)) = (y,x)$$

(a)

Find the group generated by $\alpha_1$ and $\alpha_1$. That is, find the smallest set of transformations that

• is closed under composition
• contains the identity transformation
• contains the inverses of all the transformations in the set.

(b)

Give the formulas of all the transformations in this group

(c)

Describe all the transformations in the group geometrically, by what they do to the plane

(d)

Compute the Cayley table for the group.