Name _
Date_Period_
Honors Algebra 1
Functions Test Review
For questions 1-16, Give the domain and range of each. Tell if it is a function.
1)
2)
3)
4)
D:
R:
Function?
5)
D:
R:
Function?
6)
D:
R:
Function?
9)
7)
D:
R:
Function?
10)
D:
R:
Function?
13)
D:
Math 701, Spring 2007, Exam 2
The exam ends at 1:55 PM.
Note! You must show sucient work to support your answer. Write your answers
as legibly as you can on the blank sheets of paper provided. Use only one side of
each sheet; start each problem on a new s
MATH 701 SPRING 2002
HOMEWORK 3
Due Friday, February 1 at the beginning of class.
11. Use the ideas in the proof of Cayleys Theorem to nd permutations a, b S8
which satisfy: a4 = 1, b2 = a2 , ba = a3 b, and the permutations ai bj , with
0 i 3 and 0 j 1, a
MATH 701 SPRING 2002
HOMEWORK 2
Due Friday, January 25 at the beginning of class.
4. Let n be a xed positive integer, and let C be the group C \ cfw_0 under
multiplication. How many subgroups of C have n elements? What are
they? Justify your answer.
5. (p
MATH 701 SPRING 2007
HOMEWORK 7 SOLUTION
Due Tuesday, March 27 at the beginning of class.
15. (3 points) Let I and J be ideals of the ring R which satisfy I + J = R.
R
Prove that I J R R .
=I
J
Consider the ring homomorphism : R R R , which is given by (r
MATH 701 SPRING 2007
HOMEWORK 2 SOLUTIONS
Due Tuesday, January 30 at the beginning of class.
4. Let n be a xed positive integer, and let C be the group C \ cfw_0 under
multiplication. How many subgroups of C have n elements? What are
they? Justify your an
MATH 701 SPRING 2007
HOMEWORK 9 SOLUTIONS
Due Tuesday, April 24 at the beginning of class.
26. (3 points) Let S be a multiplicatively closed subset of the commutative
ring R. Suppose that 0 S . Prove that there exists a prime ideal of R
/
which is disjoin
MATH 701 SPRING 2007
HOMEWORK 1 SOLUTIONS
Due Tuesday, January 23 at the beginning of class.
1. Complete the following multiplication table for S3 , which is generated by = (1, 2) and = (1, 2, 3). All entries should be of the
form i j with 0 i 1 and 0 j 2
Math 701, Spring 2007, Exam 1
The exam ends at 6:30 PM.
Bring your solutions to my oce 300B when you nish.
Note! You must show sucient work to support your answer. Write your answers
as legibly as you can on the blank sheets of paper provided. Use only on
Math 701, Spring 2007, Final Exam
The exam ends at 1:00 PM.
If I am not here when you nish, then please bring your solutions to my
oce 300B.
Note! You must show sucient work to support your answer. Write your answers
as legibly as you can on the blank she
Math 701, Spring 2007, Exam 2
The exam ends at 1:55 PM.
Note! You must show sucient work to support your answer. Write your answers
as legibly as you can on the blank sheets of paper provided. Use only one side of
each sheet; start each problem on a new s
MATH 701 SPRING 2002
HOMEWORK 1
Due Friday, January 18 at the beginning of class.
1. Complete the following multiplication table for S3 , which is generated by
= (1, 2) and = (1, 2, 3). All entries should be of the form i j with
0 i 1 and 0 j 2.
1 2 2
1
Math 701, Spring 1998, Exam 1
1. (17 points) Let G be a group of order pk for some prime integer p and some
integer k with 1 k . Prove that the center of G has more than one element.
(You are to reprove a result we did in class. This is a well known theor
Math 701, Spring 2000, Exam 1
Note! You must show sucient work to support your answer. Write your answers
as legibly as you can on the blank sheets of paper provided. Use only one side of
each sheet; start each problem on a new sheet of paper; and be sure
MATH 701 SPRING 2007
HOMEWORK 2
Due Tuesday, January 30 at the beginning of class.
4. Let n be a xed positive integer, and let C be the group C \ cfw_0 under
multiplication. How many subgroups of C have n elements? What are
they? Justify your answer.
5. (
Name the muscle,
A: (Action), O:
(Origin), and I:
(Insertion)
FRONTALIS - A: (Action) Elevates
eyebrows in glancing upward and
expressions of surprise or fright; draws
scalp forward and wrinkles skin of
forehead; O: (Origin) Galea aponeurotica;
I: (Insert
MATH 701 SPRING 2007
HOMEWORK 7
Due Tuesday, March 27 at the beginning of class.
15. (3 points) Let I and J be ideals of the ring R which satisfy I + J = R.
R
Prove that I J R R .
=I
J
19. (3 points) Let R be a commutative ring and let N be the set of nil
1993, Final
Problems 1, 2, and 3 are worth 15 points each. Each of the other problems is worth
16 points.
1. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTEREXAMPLE.)
If H and K are normal subgroups of the group G , with H K , then
=
GG
=K.
H
2
MATH 701 SPRING 2007
HOMEWORK 3
Due Tuesday, February 6 at the beginning of class.
7. For each positive integer n, let Gn be a group. Let
Gn be the direct product
n=1
of the groups cfw_Gn and
Gn be the coproduct of the groups cfw_Gn . For each
n=1
n, let
MATH 701
HOMEWORK 12
Due Friday, April 21 at the beginning of class.
49. Let I and J be ideals of the ring R which satisfy I + J = R. Prove that
R R
R.
I J = I
J
50. Let R be a commutative ring and let N be the set of nilpotent elements of
R. (In other w
MATH 701
HOMEWORK 11 SPRING 2002
Due Friday, April 12 at the beginning of class.
45. Let I and J be ideals of the ring R which satisfy I + J = R. Prove that
R R
R.
I J = I
J
46. Let R be a commutative ring and let N be the set of nilpotent elements of
R.
MATH 701
HOMEWORK 1
Due Friday, January 21 at the beginning of class.
4. Let n be a xed positive integer, and let C be the group C \ cfw_0 under
multiplication. How many subgroups of C have n elements? What are
they? Justify your answer.
5. (page 36, numb
MATH 701
HOMEWORK 12 SPRING 2002
Due Friday, April 19 at the beginning of class.
49. (Page 133, number 4) Show that x3 + x2 + 1 is irreducible in
Z
(2) [x]
and that
Z
[x]
(2)
3 +x2 +1)
(x
is a eld with 8 elements.
50. (Page 133, number 5) Construct elds w
MATH 701
HOMEWORK 11
Due Friday, April 14 at the beginning of class.
46. Fill in the blank with some statement about the relationship of the es and
the f s. Prove the resulting statement. Let p be a prime integer, and let G
be the Abelian group
G=
Z
Z
Z
1994, Exam 1
1. (16 points) Let K and N be normal subgroups of the group G with N K .
G
We view K = cfw_kN | k K as a subgroup of the group N = cfw_gN | g G .
N
G
It is clear that K is a normal subgroup of N . Prove that
N
G
N
K
N
and
G
K
are isomorphic.
Math 701, Spring 2002, Exam 1
Note! You must show sucient work to support your answer. Write your answers
as legibly as you can on the blank sheets of paper provided. Use only one side of
each sheet; start each problem on a new sheet of paper; and be sure
Math 701, Spring 2002, Final Exam
Note! You must show sucient work to support your answer. Write your answers
as legibly as you can on the blank sheets of paper provided. Use only one side of
each sheet; start each problem on a new sheet of paper; and be
Math 701, Spring 1998, Final Exam
Each question is worth 15 points.
1. Prove that no group of order 56 is simple.
2. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTEREXAMPLE.)
Let K and N be subgroups of the the nite group G , with |K | |N | = |
MATH 701 SPRING 2007
HOMEWORK 8
Due Tuesday, April 17 at the beginning of class.
22. (3 points) (Page 133, number 4) Show that x3 + x2 + 1 is irreducible in
Z
(2) [x]
and that
Z
[x]
(2)
3 +x2 +1)
(x
is a eld with 8 elements.
23. (3 points) (Page 133, numb
1994, Final
Each problem is worth 18 points.
1. TRUE or FALSE. (If true, PROVE it. If false, give a COUNTEREXAMPLE.)
If H and K are subgroups of the nite abelian group G , with H K , then
=
GG
=K.
H
2. Let G be a group of order p2 q , where p and q are pr