Math 30710
Exam 1
October 5, 2011
Name
This is a 50-minute exam. Books and notes are not allowed. Make sure that your work is legible, and make sure
that it is clearly marked where your answers are. S
Math 30710
Practice Exam 1
February 27, 2013
Name
This is a 50-minute exam. Books and notes are not allowed. Make sure that your work is legible, and
make sure that it is clearly marked where your ans
PRACTICE MIDTERM 1
MATH 30710
Name:
GRADES
Problem
Points
1
2
3
4
5
Total=
/50
Problem 1. Give an example (without proof) of:
(a) Maps f : A B and g : B A so that f is injective, g is surjective, and
MIDTERM 1
MATH 30710
Name:
GRADES
Problem
Points
1
2
3
4
5
Total=
/50
Problem 1. Give an example (without proof) of:
(i) a set A and a map f : A A that is injective, but not bijective.
(ii) Three grou
PRACTICE MIDTERM 1
MATH 30710
Name:
GRADES
Problem
Points
1
2
3
4
5
Total=
/50
Problem 1. Give an example (without proof) of:
(a) an isomorphism from the additive group R to the multiplicative group R
6. Cyclic Groups
21
55. Let G be cyclic and let a be a generator for G. For x, y G, there exist m, n Z such that x = am
and y = bn . Then xy = am bn = am+n = an+m = an am = yx, so G is abelian.
56. We
44
33. a. No
13. Homomorphisms
b. No
c. No
34. a. No
b. Yes
c. No
35. a. Yes, 180
b. Yes
c. No
36. a. Yes, 60 , 120 , and 180
b. Yes
37. a. Yes, 120
b. Yes
c. No
38. a. No
d. (1, 0) and (0, 1)
39. a.
CONTENTS
0. Sets and Relations
1
I. Groups and Subgroups
1.
2.
3.
4.
5.
6.
7.
Introduction and Examples 4
Binary Operations 7
Isomorphic Binary Structures 9
Groups 13
Subgroups 17
Cyclic Groups 21
Gen
Math 30710
Practice Exam 2 Answers
April 17, 2013
Name
This is a 50-minute exam. Books and notes are not allowed. Make sure that your work is legible, and
make sure that it is clearly marked where you
Math 30710
Practice Exam 2
April 17, 2013
Name
This is a 50-minute exam. Books and notes are not allowed. Make sure that your work is legible, and
make sure that it is clearly marked where your answer
30
9. Orbits, Cycles, and the Alternating Groups
Let : G G00 be defined by (a) = a1 . Clearly is one to one and maps G onto G00 . From
the equation a b = ba derived above, we have (ab) = (ab)1 = b1 a1