Spring 2006

100 points possible.

Important: For problems 7 through 11 (which are proofs), do three of the five problems. Clearly indicate which three you want me to grade (15 points each, 45  points total, no bonus credit).

Do all of problems 1 through 6.

0. (1 pt.) One free point for you today.

1. (9 pts.) Describe all group automorphisms f:Z₁₅®Z₁₅ by giving the possible values for f(1).

2. (9 pts.) Find the order of 18 in the cyclic group Z₂₄.

3. (9 pts.) Find the order of (1,1) in the group Z₁₅×Z₁₂.

4. (9 pts.) Find all subgroups of Z₁₈.

5. (9 pts.) List, up to isomorphism, all abelian groups of order 675.

6. (9 pts.) Let S₃ denote the group of all permuatations of {1,2,3}. In cycle notation,

S3={(1),(1 2),(1 3),(2 3),(1 2 3),(1 3 2)}.

Let H=á(2 3)ñ. Find all right cosets of H in S₃.

For problems 7 through 11, do three of the five problems. Clearly indicate which three you want me to grade. 15 points each, 45 points total, no bonus credit.

7. Define ~ on Z×Z as follows: "(x,y),(z,w) Î Z×Z,

(x,y) ~ (z,w)Û x+w=y+z.

Prove that ~ is an equivalence relation.

8. Let + be defined on Z×Z in the usual way: "(x,y),(z,w) Î Z×Z,

(x,y)+(z,w)=(x+z,y+w).

Define ~ on Z×Z as follows: "(x,y),(z,w) Î Z×Z,

(x,y) ~ (z,w)Û x+w=y+z.

Assume that ~ is an equivalence relation. Prove that ~ is a congruence relation with respect to +.

9. Define * on Q as follows: "r,s Î Q,

r*s=3rs-r-s+ 2 3 .

Define f:Q® Q by f(x)=3x-1. Prove that f is an isomorphism from áQ,*ñ to áQ,·ñ.

10. Let ~ be an equivalence relation on a set S. Recall that `a={x Î S | x ~ a}. Suppose c,d Î S with c ~ d. Prove that `c=`d.

Prove it directly, from the definitions, not by invoking fancy theorems. Hints: How do you prove two sets are equal? How do you prove one set is a subset of another set? What does it mean, to say that z Î `a?

11. Let áG,*ñ be a group, and let H £ G. Recall that x*H={x*h | h Î H}. Suppose a,b Î G and a^-1*b Î H. Prove that b*H=a*H.

Prove it directly, from the definitions, not by invoking fancy theorems. Hints: How do you prove two sets are equal? How do you prove one set is a subset of another set? What does it mean, to say that z Î x*H?