F15-319

Introduction to Abstract Algebra I

Fall 2015

Instructor: Kwangho Choiy

• Office: Neckers 283 (618-453-6508)

• E-mail: kchoiy_at_siu_dot_edu 

• Office Hours: Mon/Fri 12:00pm - 03:00pm, or by appointment. Emails are also available for simple questions.

Course Website: https://sites.google.com/site/kchoiy/home/teaching/previous-courses/math319

Class Meeting: MWF 11:00am - 11:50am in EGRA 322

Textbook: A First Course in Abstract Algebra 7th edition by John B. Fraleigh.

Syllabus / Course Schedule: It is required to read carefully our syllabus and schedule linked here [math319-syllabus and schedule], UPDATED on 8/20/2015.

Exams: There will be two mid-term exams and one final. No make-up exam will be accepted.

1.  Exam 1 (in class on 9/28, Mon) covers Sec 0--5.

2.  Exam 2 (in class on 10/28, Wed) covers Sec 6, 8, 9, 10.

3.  Final Exam (10:15am-12:15pm on Dec 18, Fri) will be comprehensive: approximately, 40% on Sections 0--6, 8--10 and 60% on Sections 11, 13--19 (excluding 20, 21).  (**See the sheet of practice problems of Final, which contains all details**)

Each HOMEWORK ASSIGNMENT will be posted as below at least one week ahead of the due date (see the Course Schedule above for a tentative assignment schedule!):

*HW Policies: You should show all your work and submit it in class on the due date. No late homework will be accepted.*

1. HW 1 (due on Sep. 4, Fri)

2. HW 2 (due on Sep. 16, Wed)

3. HW 3 (due on Sep. 23, Wed)

4. HW 4 (due on Oct. 9, Fri)

5. HW 5 (due on Oct. 16, Fri)

6. HW 6 (due on Oct. 23, Fri)

7. HW 7 (due on Nov. 9, Mon)

8. HW 8 (due on Nov. 18, Wed)

9. HW 9 (due on Dec. 2, Wed)

10. HW 10 (due on Dec. 9, Wed)

updates and remarks - math319 - Fall 2015:

[Dec 11, Fri] Lecture 43: Sec 19 -- proof of the fact that a finite ID is a field, def of characteristic of a ring, some theorem and examples related to the characteristic. HW 10 returned with solution. 

[Dec 9, Wed] Lecture 42: Sec 18, 19 -- properties of a ring, some arguments about zero divisors, integral domain, field.

[Dec 7, Mon] Lecture 41: Sec 18, 19 -- zero divisors, integral domain, a diagram having all new terms related to a ring with examples. The practice problems for Final distributed! -- read carefully.

[Dec 4, Fri] Lecture 40: Sec 18 -- def of a ring, motivation, examples, new terms related to a ring. HW 9 returned with solution.

[Dec 2, Wed] Lecture 39: Sec 17 -- Burnside's Theorem and a couple of examples. HW 10 (due on Dec. 9, Wed) distributed!

[Nov 30, Mon] Lecture 38: Sec 16 -- a bijection between Gx and the collection of left cosets of G_x in G, some corollaries related to counting issues. 

[Nov 23, Mon] Lecture 37: Sec 16 -- introducing new terms related to group actions, examples, and properties. HW 8 returned with solution! 

[Nov 20, Fri] Lecture 36: Sec 16 -- definition of group actions, examples. HW 9 (due on Dec. 2, Wed) distributed! 

[Nov 18, Wed] Lecture 35: Sec 15 -- discussions on the definition of the commutator subgroup, properties of the commutator subgroup and their proof, some other terms. / Sec 16 -- motivation of the def of group actions.

[Nov 16, Mon] Lecture 34: Sec 15 -- introduce the commutator subgroup C(G) = [G,G] of G and examples, refer to this link to see that the set {xyx^{-1}y^{-1} : x, y in G} does not always form a subgroup, since the set is not closed (although the set always contains the identity and the inverse element of any xyx^{-1}y^{-1}. According to the link, the first finite group that this set does not form a subgroup is of order 96. Thus, we need to define C(G) as the subgroup generated by the set {xyx^{-1}y^{-1} : x, y in G}, or the smallest subgroup containing the set, or the set {a_1a_2...a_n : a_i's are of the forms xyx^{-1}y^{-1}. Notice the last set now becomes a subgroup, since it is closed.

[Nov 13, Fri] Lecture 33: Sec 15 -- more details about the solvability of polynomial and A_n, n is greater or equal to 5, behavior of normal subgroup via homomorphisms, introduce the center Z(G) of a group G and its properties. HW 7 returned with solution.

[Nov 9, Mon] Lecture 32: Sec 15 -- how to simplify a given complicated factor group using the fundamental theorem of homomorphisms, def of simple groups, examples, a digressive argument relating a simple group An to the solvability of polynomial (if more interested, see this link).   HW 8 (due on Nov. 18, Wed) distributed!

[Nov 6, Fri] Lecture 31: Sec 14 -- fundamental theorem of homomorphisms, some examples.

[Nov 4, Wed] Lecture 30: Sec 14 -- normal subgroup, examples for normal subgroup, non-normal, "H is normal in G <==> aHbH=abH is well-defined", if H is normal (then gHg^-1 = H for all g in G), then the collection of cosets (=left cosets=right cosets) forms a group with the group law "aHbH=abH", called the factor group of G by H, denoted by G/H.

[Nov 2, Mon] Lecture 29: Sec 13 -- more about ker phi, Sec 14 -- an example for which condition is required to give a group structure to the collection of left (right) cosets. HW 7 (due on Nov. 9, Mon) distributed!

[Oct 30, Fri] Lecture 28: Sec 13 -- recall def of homomorphism from G to G', examples, properties, introduce the kernel ker phi. EXAM 2 returned with solution.

[Oct 28, Wed] Lecture 27: EXAM 2

[Oct 26, Mon] Lecture 26: Review for EXAM 2, HW6 returned with solution.

[Oct 23, Fri] Lecture 25: Sec 11 -- more applications for finitely generated abelian group, sketch of proof of gcd (n,m)=1 iff Zn x Zm isomorphic to Znm.

[Oct 21, Wed] Lecture 24: Sec 11 -- classification of finite abelian groups, finitely generated abelian group, applications, examples, def of the direct product. The practice problems for EXAM 2 distributed!

[Oct 19, Mon] Lecture 23: Sec 10 -- cosets, Lagrange theorem, its applications, index. HW 5 returned with solution.

[Oct 16, Fri] Lecture 22: Sec 9 -- transpositions, An the alternating group on n letters. HW 6 (due on Oct. 23, Fri) distributed!

[Oct 14, Wed] Lecture 21: Sec 9 -- defs and examples for orbits, cycles. HW4 returned (the solution will be distributed on Fri, Oct. 16). 

[Oct 9, Fri] Lecture 20: Sec 8 -- Cayley theorem, examples, proof. HW 5 (due on Oct. 16, Fri) distributed!

[Oct 7, Fri] Lecture 19: Sec 8 -- def permutations of a set A, show S_A is a group, introduce new terms and notation including S_n, D_n

[Oct 5, Mon] Lecture 18: Sec 6 -- cyclic is isomorphic to Z or Z_n and proof.

[Oct 2, Fri] Lecture 17: Sec 6 -- some relation between two subgroups of cyclic. HW 4 (due on Oct. 9, Fri) distributed! EXAM 1 returned with solution, discussions on Exam 1.

[Sep 30, Wed] Lecture 16: Sec 6 -- structure of cyclic groups, subgroup of cyclic is cyclic, gcd.

[Sep 28, Mon] Lecture 15: EXAM 1

[Sep 25, Fri] Lecture 14: Review for EXAM 1, HW3 returned with solution.

[Sep 23, Wed] Lecture 13: Sec 5 -- proof of a theorem related to cyclic subgroups / Sec 6 -- def of cyclic groups and examples, any cyclic is abelian.

[Sep 21, Mon] Lecture 12: Sec 5 -- proof of the two theorems (useful criteria to check if it is a subgroup) in Lecture 11, introduce V_4, cyclic groups, the practice problems for EXAM 1 distributed!

[Sep 18, Fri] Lecture 11: Sec 5 -- order of element, order of group, def of subgroup, examples, state two theorems to check whether a given subset is a subgroup, HW2 returned with solution.

[Sep 16, Wed] Lecture 10: Sec 4 -- elementary properties of groups. HW 3 (due on Sep. 23, Wed) distributed!

[Sep 14, Mon] Lecture 9: Sec 4 -- examples of groups, elementary properties of groups 

[Sep 11, Fri] Lecture 8: Sec 3 -- introduce "an" identity element, show the uniqueness if exists, its preservation via ~ / Sec 4 -- definition of groups

[Sep 9, Wed] Lecture 7: Sec 3 -- how to show <S, *> ~ <S', *`> e.g., Z ~ 2Z, how to show <S, *> not ~ <S', *`>, state what is preserved via ~. HW1 returned with solution, HW2 (due on Sep. 16, Wed) distributed!

[Sep 4, Fri] Lecture 6: Sec 2 -- commutative, associative binary operations, Cayley table / Sec 3 -- binary algebraic structures, def of isomorphism

[Sep 2, Wed] Lecture 5: Sec 2 -- numbers from N through C, def of a binary operation, examples, <Zn, +n> 

[Aug 31, Mon] Lecture 4: Sec 1 -- complex numbers C, modulus, argument, U_n in C, some algebraic structure in U_n.

[Aug 28, Fri] Lecture 3: Sec 0 -- more about partition, equivalence relation, congruence. HW 1 (due on Sep. 4, Fri) distributed!

[Aug 26, Wed] Lecture 2: Sec 0 -- more details about cardinality, partition, equivalence relation.

[Aug 24, Mon] Lecture 1: syllabus / Sec 0 -- set, relation on a set, equivalence relation, partition, cardinality / rough concept of groups.

[Aug 23, Sun] Supplementary Material: Visit the link to see mathematical proofs, notation, arguments, etc.