First Midterm
Dave Bayer, Modern Algebra, October 6, 1997
[1] Give an example of a group G and a subgroup H , where
(a) H is normal. What is the quotient group G/H ?
(b) H is not normal. Show that H is not normal, by nding an element g G with
the property
Second midterm
Dave Bayer, Modern Algebra, November 9, 1997
Please solve 5 of the following 6 problems. Each problem is worth 5 points for
a total of 25 points. I will also award up to 5 bonus points (recorded separately) for
particularly impressive exami
Second midterm
Dave Bayer, Modern Algebra, November 18, 1998
[1] Find the multiplicative inverse of 23 mod 103.
[2] Prove ONE of the following two assertions:
(a) Let V and W be subspaces of a vector space U . Then
dim(V ) + dim(W )
dim(V W ) + dim(V + W
Solutions to rst midterm
Dave Bayer, Modern Algebra, Exam date October 14, 1998
[1] Let G = S3 = cfw_ (), (1 2), (1 3), (2 3), (1 2 3), (1 3 2) be the symmetric group of all permutations of
cfw_ 1, 2, 3 , and let H be the subgroup H = cfw_ ( ), (1 2) G.
Solutions to second midterm
Dave Bayer, Modern Algebra, Exam date November, 1998
[1] Find the multiplicative inverse of 23 mod 103.
Solution: By the extended euclidean algorithm, we compute
103
1
23
11
0
0
which we interpret as 23 = 0 103 + 1 23
11 =
First midterm
Dave Bayer, Modern Algebra, October 14, 1998
Work as many parts of each problem as you can, while budgeting your time. For a successful
exam, it isnt necessary to answer every part of every question.
[1] Let G = S3 = cfw_ (), (1 2), (1 3), (
Final Examination
Dave Bayer, Modern Algebra, December 23, 1997
Each problem is worth 5 points for a total of 50 points. Work as much of each problem as you
can.
A
B C
D E F
G H I J
K L M N P
Q R S T U V
Figure 1
[1] Let the dihedral group D3 of symmetrie
Practice problems for second midterm
Dave Bayer, Modern Algebra, November 5, 1997
There will be two review sessions in 528 Mathematics: Friday, November 7 at
2:40pm, and Sunday, November 9 at 2:40pm. (We will move if we need more space.)
[1] Dene an abstr
Practice problems for rst midterm
Dave Bayer, Modern Algebra, September 29, 1997
[1] Give an example of a group G and a subgroup H, where
(a) H is normal. What is the quotient group G/H?
(b) H is not normal. Show that H is not normal, by nding an element