Winter 2009
Math 314
Modern Algebra

Instructor: Sergio R. Lopez, 315E Morton Hall (593-1262), lopez at math.ohiou.edu

Click here to see a copy of the First Day Handout
  
TENTATIVE SCHEDULE. 
What you see here is an ever-mutating schedule.  At the beginning of the quarter it will be identical to last quarter's schedule.  We will update it daily to reflect what we accomplished in class. I will also introduce modifications as I plan ahead but, plese, never take any information about the future very seriously.  On the other hand, information about the past should be very accurate.

The final exam will be on  Thursday, March 19, at 10:10 a.m.

Beware of it 
 
Monday
Tuesday
Wednesday
Thursday
Week 1
(January 5-8)
Section 1.1: Sets, elements, empty set. 
 Suggestions how to study
Read as much as you can out of Section 1.1.
In particular, read about subsets and the power set.
HWK: 1-3, page 10
Section 1.1: Equality of Sets, Unions and Intersections.
Read about complements and  do HWK: 4-9, page 10
Complements, sets of numbers.  DeMorgan's laws.
Double Inclusion
 HWK: 10-14, page 10
Section 1.1: Disjoint Sets, Partitions.
Section 1.2: Cartesian Products. 		HWK: 15-20, page 11 and problem 1 from page 20.
Week 2
(January 12-15)
Section 1.2:  Functions, Domain, Codomain, Range, Image, equality of functions HWK: 2-3, pages 20-21.
Section 1.2: Image of a subset, Inverse image of a subset. One-to-one, onto, One-to-one correspondences. HWK: 4-14 in pages 21-22
Section 1.2: Composition of maps and Section 1.3 HWK: 15, 16, 20-22 (page 22) and 1-12, page 26.
 
Section 1.4 HWK: 1-10, pages 31 and 32.

Week 3
 (January 19-22)
 Martin Luther King, Jr. Day
NO CLASS
Section 1.5 HWK: 1-11, page 36.
One-to-one if and only if left invertible. Onto if and only if right invertible
Section  1.6 HWK: 1-6, 8-11, page 46.
Week 4
(January 26-29)

 
Section 1.7.
Every Partition induces and equivalence relation. Every equivalence relation induces a partition.
 HWK:  1, 2 (a-d), 3-7, 10, 12-14. page 52-53.
REVIEW NO CLASS TODAY.
OU CLOSED DUE TO SNOW!
 PARTIAL EXAM I
Week 5
(February 2-5)
Section 2.6 and Section 3.1. Definition and examples of  groups. HWK: 1-10. Page 94.
HWK: 1-25  (pages. 127-130).
Theorem 3.4: Properties of group elements. HWK:
27, 28 and 30-35,
 Theorem 3.5: Characterizations of groups.
HWK: 38-41, and
45-49 (page 130)
Theorem 3.7: Generalized Associative Law.
Example 10 (abstract version)
Talk about the posted problems.

 Relationship between Theorems 3.4(e) and 3.5 and one-to-one and onto functions.
HWK: 1-4, page 138
  
Week 6
(February 9-12)
Section 3.2: definition of subgroup. Examples and discussion.  Is it the same identity?  Do they need to be the same inverses?
Theorems 3.9  and 3.10. Definition 3.11 and Theorem 3.12.		 HWK: 8,9, 11 and 17 (pages 138 and 139)
Section 3.3: Cyclic Groups. Theorem 3.15 (Infinite Cyclic Groups.)

  HWK: 18(d), 21, 23-25 page 140 and 1-7 (pages 147-8)
Theorem 3.17 (Finite Cyclic Groups).
HWK: 1-4, 6-7, 9, 19-20(page 148)
Section 3.4: Isomorphisms.

 HWK: 1-4, (page 156.)
HWK: 1-4 (pages 160-1).

Week 7
(February 16-19)
Theorem 3.20 (Subgroups of a Cyclic group.)  Theorems 3.26 and 3.28: Images of the identity and inverses.

REVIEW
PARTIAL EXAM II
Section 3.5:
The kernel and range of a homomorphisms are subgroups, respectively, of the domain and co-domain. Talk about monomorphisms, and epimorphisms.		 HWK: 25, 28, 29 (page 157). Also, 11, 12, 15, 16(a) (page 163.)
Week 8
(February 23-26)
Section 4.1: Permutations, Cyclic permutations, transpositions, decomposition of permutations as products of disjoint cycles, decomposition of cycles as products of transpositions, even and odd permutations.
 HWK: 1-4 (pages 174-5)
Section 4.1: Alternating group, Conjugates. HWK: 5-11,14-16, and 21 (page 175.) For more information about problem 21, follow this link: Conjugacy is an equivalence relation
Section 4.2: Cayley's Theorem HWK: 1-3 (page 178)
Further discussion of Cayley's Theorem. HWK: 5-7 (page 178).
Week 9
(March 2-5)
Section 4.4: Left Cosets.  HWK: Read Definition 4.8 and Theorem 4.9. Become acquainted with the concept of right cosets (analogous to the left cosets discussed in class) Make sure you can prove all statements in Theorem 4.9.  Do problems 1-3 (pages 193-4)
Section 4.4: Lagrange's Theorem and related results (Lemma 4.11, Theorem 4.13, Theorem 4.16.
HWK: 6-8, 10-11 (page 194.)
left cosets aH and bH are equal if and only if b^{-1}a belongs to H. Exploring the possibility of the sets of left cosets and of right cosets to be groups with the operation induced on subsets.  Informal introduction to normal subgroups. HWK:  7, 14-16 (page 194.)
Definition and characterizations of a Normal Subgroup. Theorem 4.18.
Definition of Quotient Group.
HWK: 8, 9, 10, 22, 32, 34 (pages 194-5)
The first poster project is due today!!!
Week 10
(March 9-12)
Section 4.5: Theorems 4.21 and  Theorem 4.23. Theorem 4.24 and Theorem 4.25, Theorem 4.26. HWK:
30-38 (page 195) and 2-4 (page 202)

HWK: 10, 11, 14 (page 203)



epi-mono factorization

REVIEW
REVIEW

The second poster project is due today!!!
FINALS WEEK
(March 16-20)



The final exam will be on  Thursday, March 19, at 10:10 a.m.