Professor:
Dr. Janet Vassilev
Office: Humanities 467
Office
Hours: MWF 11 am12 pm and by appointment.
Telephone: (505)
2772214
email: jvassil@math.unm.edu
webpage:
http://www.math.unm.edu/~jvassil
Date

Section

Topic

Homework

8/23

2

Binary Operations^{} 

8/25

3

Isomorphic
Structures

Section 2: 3, 5, 6, 9, 12, 24, 26, 28, 36
Section 3: 1, 3, 4, 12, 27

8/27

4

Groups


8/30

4

Groups


9/1

4, 5

Groups and
Subgroups

Section 4: 1, 3, 5, 7, *12, *18, 19, 20, 24, 25, 29,
31, 32, 35

9/3

5

Subgroups


9/8

5, 6

Division
Algorithm and Euclidean Algorithm

Section 5: 2, 4, 6, 14, 15, 20, 28, 31, 36, 39, 41,
43, 45, 47, 51, 54

9/10

6

Cyclic groups_{} 

9/13

6

Cyclic Groups


9/15

6,7

Cyclic groups
and Finitely generated groups

Section 6: 3,
7, 11, 18, 20, 22, 32, 33, 36, 46, 49, 55
Section 7: 4,
5, 7, 12, 13

9/17

7,8

Finitely generated
groups and dihedral groups


9/20

8

Dihedral groups


9/22

8

Groups of
permutations

Section 8: 1,
4, 6, 16, 30, 33, 35, 36, 52

9/24


Review


9/27


Midterm 1


9/29

9

Orbits, cycles
and alternating groups

Section 8: 40,
41, 46, 49
Section 9: 1,
3, 7, 8, 13, 14, 16

10/1

9

Orbits, cycles
and alternating groups

Quiz 15: What
is the orbit of 3 for the element
1 2 3 4 5 6 7 8
2 1 4 6 7 8 3 5
?

10/4

10

Cosets and
Lagrange

Quiz 16: Write out
a=(1234)(567) as a product of transpositions and determine if a is odd or
even.

10/6

10

Cosets and
Lagrange

Quiz 17:
Determine the left cosets of <4> in Z_{12}
Section 9: 23, 29, 33, 34
Section 10: 1,
4, 6, 7, 20, 26, 27, 28, 30, 32

10/8

11

Direct Products
and Finitely generated abelian groups

Quiz 18: What
are the possible orders of subgroups in a group of order 20?

10/11

11

Direct Products
and Finitely generated abelian groups

Quiz 19: Give
the definition of a direct product of groups (G_{i}, *_{i}
)give the binary operation associated to this new group.

10/13

12

Plane
Isometries

Quiz 20: Is Z_{2}
x Z_{10} isomorphic to Z_{20}?
Section 10: 12, 14, 15, 19, 34, 39, 45
Section 11: 3,
6, 11, 14, 16, 32, 36, 46, 47, 49

10/18

13

Homomorphisms

Quiz 21: List
two plane isometries of infinite order

10/20

14

Factor Groups

Quiz 22: Give
the definition of the kernel of a homomorphism.
Section 11: 50, 51
Section 13: 2, 4,
5, 8, 9, 17, 20, 23, 32, 37, 39, 44, 49, 50

10/22

14

Factor Groups

Quiz 23: Give
an example of a normal subgroup in a nonabelian group.

10/25

15

More on Factor
Groups

Quiz 24: What
is a criterion on H for G/H to be a group?

10/27

16

Group Actions

Quiz 25: Let G be Z_{2} x Z_{6} and
H=<(1,3)>. What is the order of
G/H?
Section 14: 2, 6, 10, 14, 22, 23, 30, 31, 34
Section 15: 4,
12, 19, 34, 35, 36

10/29

16, 17

Group Actions
and Counting

Quiz 26: Give the
definition of a group acting on a set.

11/1

17

Group Actions
and Counting

Quiz 27: Give
an example of a group which acts on the diagonals of a square.

11/3

18

Rings

Quiz 28: State
Burnside’s Theorem
Section 16: 1, 3, 8, 11, 12, 14
Section 17: 1,
2, 4

11/5


Review

Quiz 29: Give
the definition of a ring

11/8


Midterm 2


11/10

18

Rings and
Fields

Section 18: 2,
6, 8, 10, 16, 18, 21, 24, 33, 37, 38, 39, 44, 46, 49 (due next Friday)

11/12

19

Integral Domains

Quiz 31: What
is a ring homomorphism?

11/15

20

Fermat’s and
Euler’s Theorems

Section 19: 6,
10, 17, 23, 25, 29
Section 20: 4, 7, 10, 12, 14, 23, 27
Quiz 32: Give
an example of a zero divisor and the ring it lives in.

11/17

21

Field of
Quotients

Quiz 33: What
is phi(100)?

11/19

22

Polynomial
Rings

Quiz 34: What
is the set we used to construct the field of fractions of a domain D?

11/22

22

Polynomial
Rings

Quiz 35: What
is the difference between a power series and a polynomial?

11/24

23

Factorization
of Polynomials

Quiz 36: If R
is a domain, is R[x] a domain?

11/29

23

Factorization
of Polynomials

Quiz 37: If a
is a zero of f(x) then tell me a factor of f(x).

12/1

24, 26

Noncommutative
rings and Homomorphisms and Factor
Rings

Quiz 38: State Gauss Lemma.
Section 21: 1, 4, 12
Section 22 5, 8, 11, 12, 23, 27, 28
Section 23: 2,
9, 10, 12, 14, 19, 25, 35

12/3

27

Prime and
Maximal Ideals


12/6

28

Groebner Bases


12/8


Review


12/10


Review


12/17


Final exam

