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. Show all work. Good luck!
1. (10 points) Give an example
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 answers are. Show all work. Good luck!
1. (10 points) Give
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 g f is
neither injective nor surjective;
(b) two infini
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 groups of order 8 that are pairwise non-isomorphic.
(iii) A
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>0 ;
(b) a infinite non abelian group;
(c) an Abelian g
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 can show it if G is abelian. Let a, b G so that an , b
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. Yes, 90 and 180
b. Yes
b. No
c. Yes
c. No
c. No
40. 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
Generators and Cayley Digraphs 24
II. Permutations, Cosets
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 your answers are. Show all work. Good luck! (Note the
poss
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 answers are. Show all work. Good luck! (Note the
possible poi
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 = a1 b1 = (a)(b),
which is the homomorphism property f