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.
Exam 1 (in class on 9/22, Fri) covers Chapters 1.1-1.7; 2.1-2.4.
Exam 2 (in class on 10/27, Fri) covers Chapters 3.1 -3.5; 4.1-4.3, 5.5 - prac, partial sol
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.*
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).
HW 2 (due on Sep. 11, Mon)
HW 3 (due on Sep. 18, Mon)
HW 4 (due on Oct. 6, Fri) - ** Typo in #4, K |> HK must be corrected to K <| HK.
HW 5 (due on Oct. 16, Mon) - ** Ignore #1
HW 6 (due on Oct. 23, Mon)
HW 7 (due on Nov. 8, Wed) - ** Extended to Nov. 10th.
HW 8 (due on Nov. 20, Mon) - ** No deed to submit #8, #9. / **Fixed typos in #1,#4.
HW 9 (due on Dec. 1, Fri) - **Extended to Dec. 6th.
Special Assignment (due on Dec. 4, Mon)- **voluntary.
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.