17fall_519

Algebraic Structures I

- Fall 2017 -

Instructor: Kwangho Choiy

• Office: Neckers 283 (618-453-6508)

• E-mail: kchoiy_at_siu_dot_edu 

• Office Hours: Mon/Wed/Fri 11am-noon; Wed 1-4pm, or by appointment. Emails are also available.

Course Website: https://sites.google.com/site/kchoiy/home/teaching/f17_519

Class Meeting: MWF 12:00pm - 12:50pm in Neckers 218.

Textbook: Abstract Algebra 3rd edition by David S. Dummit and Richard M. Foote. 

Course Objectives/Topics: Our goal is to learn the theory of groups and rings. The topics include: structure of groups, subgroups, quotient groups, isomorphism theorems, group actions, Sylow theorems, semi-direct product of groups, classification of finite groups, structure of rings, ideals, UFD, PID, Euclidean domain, polynomial rings, irreducibility criteria.

Syllabus / Course Schedule: It is required to read carefully our syllabus and schedule linked [here], updated on 8/27/2017.

Exams: There will be two mid-term exams and one final in the classroom.

1.  Exam 1 (in class on 9/22, Fri) covers Chapters 1.1-1.7; 2.1-2.4.

2.  Exam 2 (in class on 10/27, Fri) covers Chapters 3.1 -3.5; 4.1-4.3, 5.5 - prac, partial sol

3.  Final (12:30pm-2:30pm on 12/11, Mon) covers Chapters 7.1-7.4, 7.6; 8.1-8.3; 9.1-9.5 - prac

Each HOMEWORK ASSIGNMENT will be posted as below at least one week ahead of the due date (see the Course Schedule linked 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. 1, Fri) - **Do not need to do questions of Chapter 1.2 in #3 of HW 1 (updated on 8/30); ** Extended to NOON on Sep. 5th, Tue (updated on 9/1).

2. HW 2 (due on Sep. 11, Mon)

3. HW 3 (due on Sep. 18, Mon)

4. HW 4 (due on Oct. 6, Fri) - ** Typo in #4, K |> HK must be corrected to K <| HK.

5. HW 5 (due on Oct. 16, Mon) - ** Ignore #1

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

7. HW 7 (due on Nov. 8, Wed) - ** Extended to Nov. 10th.

8. HW 8 (due on Nov. 20, Mon) - ** No deed to submit #8, #9. / **Fixed typos in #1,#4.

9. HW 9 (due on Dec. 1, Fri) - **Extended to Dec. 6th.

10. Special Assignment (due on Dec. 4, Mon)- **voluntary.

11. HW 10 (no need to submit)

Updates and Remarks - MATH519 - Fall 2017:

[Dec 8, 2017] Lecture 43:  Chapter 9 - sketch proof of R[x]/I[x] isom to (R/I)[x], example, Eisenstein Criterion for Z and for ID and application, ideas for the proof of UFD[x] being UFD, its applications, Gauss Lemma, and further arguments related to factorizations in R[x] and Quot(R)[x].

   -- Graded HW 9 and Special Assignment are returned.

[Dec 6, 2017] Lecture 42:  Chapter 9 - definition of polynomial rings, properties, examples, UFD[x] is UFD, F[x] is ED with sketch of proof.

   -- Solution of HW 9 is handed out.

[Dec 4, 2017] Lecture 41:  Chapter 8 - more discussions related to prime/irreducible elements; UFD,PID,ED, more examples.

   -- A PDF file of practice problems, selected topics, and details for FINAL is E-mailed.

[Dec 1, 2017] Lecture 40:  Chapter 8 - Z[sqrt{-5}] is not UFD, Examples in factorization in Z[i], more arguments related to prime/irreducible elements, Discussed questions in HW 9.

[Nov 29, 2017] Lecture 39:  Chapter 8 - proved PID is UFD, introduced ACC, Some arguments related to prime/irreducible elements, gave partial solution for HW 9 questions.

[Nov 27, 2017] Lecture 38:  Chapter 8 - prime element, irreducible element, examples, outlined their properties/relations.

   -- Graded HW 8 with solution is returned

[Nov 20, 2017] Lecture 37:  Chapter 8 - proved PID is UFD, an example for UFD, not PID with proof, detailed about the notion of uniqueness.

   -- HW 9 (due on Dec. 1, Fri) is handed out.

   -- Special Assignment (due on Dec. 4, Mon; voluntary to submit) is E-mailed.

[Nov 17, 2017] Lecture 36:  Chapter 8 - further concrete examples for ED, relationship between 3 properties and ED,PID,UFD, defined UFD, example for PID, proof of Z being PID.

[Nov 15, 2017] Lecture 35:  Chapter 8 - discussed ED => PID => UFD but not the other directions, concrete examples for ED.

[Nov 13, 2017] Lecture 34:  Chapter 8 - 3 main properties integers satisfy, introduced how to expand such properties to abstract ring-related notions, introduced ED, PID, UFD, defined ED, PID, examples.

   -- Graded HW 7 with solution is returned

[Nov 11, 2017] An additional note for Chinese Remainder Theorem is E-mailed.

[Nov 10, 2017] Lecture 33:  Chapter 7 - motivating question for Chinese Remainder Thm, comaximal, Proof of Chinese Remainder Thm, remarks, examples.

   -- HW 8 (due on Nov. 20, Mon) is handed out.

[Nov 8, 2017] Lecture 32:  Chapter 7 - some properties of ideals, principal/prime/maximal ideals, their properties related to ID, field, examples.

[Nov 6, 2017] Lecture 31:  Chapter 7 - more about kernel, ideals, factor rings, examples, isomorphism theorem.

[Nov 3, 2017] Lecture 30:  Chapter 7 - properties of rings, examples, ring homomorphism and its properties, kernel of the ring homomorphism, ideal. 

   -- HW 7 deadline is now Nov. 10th.

[Nov 1, 2017] Lecture 29:  Chapter 7 - introduced several kinds of rings, examples, diagram. 

   -- Graded Exam 2 is returned // HW 7 (due on Nov. 8, Wed) is handed out.

[Oct 30, 2017] Lecture 28:  Chapter 7 - introduced a ring, definitions, examples, new terms.

[Oct 27, 2017] Lecture 27: Exam 2 taken.

   -- Exam 2 solution is handed-out.

[Oct 25, 2017] Lecture 26:  Chapter 5.2 - fundamental thm for finite ab gp; f.g. abelian gp, remarks, examples, some useful arguments.

   -- A PDF file of Partial Solutions for practice problems for EXAM 2 is E-mailed // An additional note for further topics for group theory is E-mailed. // Graded HW 6 with solution is returned

[Oct 23, 2017] Lecture 25:  Chapter 5.5 - recall the def of semi-direct products, remarks, examples.

   -- A PDF file of practice problems, selected topics, and details for EXAM 2 is E-mailed.

[Oct 20, 2017] Lecture 24:  Chapter 4 - introduced further topics in group theory, three special group actions, conjugacy classes, class equation, examples / Chapter 5.5 - define a semi-direct product.

[Oct 18, 2017] Lecture 23:  Chapter 4.5 - proofs of 3 Sylow thms, remarks from the proofs. 

   -- Graded HW 5 with solution is returned

[Oct 16, 2017] Lecture 22:  Chapter 4.5 - ideas of 3 Sylow thms, two lemmas and proofs.  

   -- HW 6 (due on Oct. 23, Mon) is handed out.

[Oct 13, 2017] Lecture 21:  Chapter 4 - outline of topics in the chapter, a relationship between cyclic, abelian, solvable, non-abelian / Chapter 4.5 - statements of three Sylow theorems, Sylow-p-subgroup, n_p=1 <=> a Sylow-p-subgroup is normal in G.

[Oct 11, 2017] Lecture 20:  Chapters 3.4/3.5 - more arguments about subnormal, normal series, composition series, simple, solvable group, more examples, discussed: Cauchy Thm, |G|=p, prime, <=> Z_p <=>simple and abelian finite group, A_n with sgn, A_n, n >=5, is simple. 

   -- Graded HW 4 with solution is returned // An additional note for series of subgroups is haded-out.

[Oct 6, 2017] Lecture 19:  Chapter 3.4 - defined subnormal, normal series, composition series, examples, simple, solvable group. 

   -- HW 5 (due on Oct. 13, Fri) is handed out. // An additional note for series of subgroups is E-mailed.

[Oct 4, 2017] Lecture 18:  Chapter 3.3 - proofs of the 3 iso thms, remarks.

[Oct 2, 2017] Lecture 17:  Chapter 3.3 - a motivating example for isomorphism thms, introduced 3 iso thms and relevant remarks.

[Sep 29, 2017] Lecture 16:  Chapters 3.2-3.3 - Lagrange Theorem, examples, introduced the factor group, binary operation on the collection of left (right) cosets of H in G <=> H is normal in G, example for which H is not normal and the operation is not well-defined. 

   -- HW 4 (due on Oct. 6, Fri) is handed out.

[Sep 27, 2017] Lecture 15:  Chapters 3.1-3.2 - sketch of proofs of several equivalent arguments for being normal, recalled the definition/argument of equivalence relations, introduced left/right cosets, [G:H] = # of left cosets = # of right cosets. 

   -- Graded Exam 1 with solution is returned.

[Sep 25, 2017] Lecture 14: Chapter 2.5 - the lattice of subgroups, examples / Chapter 3 - introduced objecctives / Chapter 3.1 - recall def of homomorphism, properties, study kernel, definition of normal subgroup, equivalent conditions of being normal, examples, motive of cosets, factor groups

[Sep 22, 2017] Lecture 13: Exam 1 taken.

[Sep 20, 2017] Lecture 12: Chapter 2.3 - proved some of the main properties, discussed structure of cyclic groups.  

   -- Graded HW 3 with solution is returned.

[Sep 18, 2017] Lecture 11: Chapter 2.3 - defined cyclic groups, generator, examples, properties. 

   -- An additional note for cyclic groups is E-mailed.

[Sep 15, 2017] Lecture 10: Chapter 2.2 - examples related to inclusions {2} =< Z(G) =< C_G(A) =< N_G(A) =< G, defined commutator subgroup, stabilizer, examples, properties.

[Sep 13, 2017] Lecture 9: Chapter 2.2 - centralizer, normalizer, center, examples, their properties. / Chapter 2.4 - <A> the subgroup generated by a subset A in a group G, examples. 

   -- Graded HW 2 with solution is returned.

[Sep 11, 2017] Lecture 8: Chapter 2.1 - def of subgroups, examples, arguments related to subgroups, subgroup criterion and proof. / Chapter 2.2 - introduced the centralizer. 

   -- HW 3  (due on Sep. 18, Mon) is handed out.

[Sep 8, 2017] Lecture 7: Chapter 1.7 - definition, motivation of group actions, examples,  several terms of group actions.

[Sep 6, 2017] Lecture 6: Chapters 1.2-1.5 - introduced Q_8 - the quaternion group of order 8 / Chapter 1.6 - definitions of homomorphism, isomorphism, examples, properties. 

   -- Graded HW 1 with solution is returned. // HW 2 is distributed in a paper form.

[Sep 1, 2017] Lecture 5: Chapters 1.2-1.5 - some properties on S_n, even, odd, permutations, D_{2n}, its group presentation, examples. 

   -- HW 1 Due is extended // HW 2  (due on Sep. 11, Mon) is emailed. 

[Aug 30, 2017] Lecture 4: Chapters 1.2-1.5 - more about S_n, permutations, cycles, transpositions, orbits, examples, S_A with finite set A.

[Aug 28, 2017] Lecture 3: Chapter 1.1 - abelian groups, more properties of a group, order of an element in a group, order of a group / Chapters 1.2-1.5 - introduced S_n the symmetric group of degree n. 

[Aug 27, 2017] Office Hours Updated as above!

[Aug 25, 2017] Lecture 2: Chapter 1.1 - binary operation, associative, commutative, examples, Z, Q, R, C, Z_n. 

   -- HW 1 (due on Sep. 1, Fri) is distributed.

[Aug 23, 2017] Lecture 1: Syllabus distributed / listed course objectives and topics / Chapter 1.1 - definition of groups, binary operation, examples

[Aug 18, 2017] Syllabus is uploaded above in PDF.