F16-519

Algebraic Structures I

Fall 2016

Office: Neckers 283 (618-453-6508)
E-mail: kchoiy_at_siu_dot_edu
Office Hours: Mon/Wed 2:00pm-5:00pm, or by appointment. Emails are also available.

Course Website: https://sites.google.com/site/kchoiy/home/teaching/previous-courses/519-f16

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

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

Syllabus / Course Schedule: It is required to read carefully our syllabus and schedule linked [here], updated on 8/16/2016.

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

Exam 1 (in class on 9/23, Fri) covers Chapters 1.1-1.7 and 2.1-2.3.
Exam 2 (in class on 10/28, Fri) covers Chapers 3.1-3.5 and 4.1-4.4.
Final (12:30pm-2:30pm on 12/12, Mon) covers Chapters 4.5, 5.2, 5.5, 6.1, 7.1-7.6, 8.1-8.3.

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 9/2, Fri) - updated on 8/29.
HW 2 (due on 9/12, Mon)
HW 3 (due on 9/19, Mon)
HW 4 (due on 10/7, Fri)
HW 5 (due on 10/17, Mon)
HW 6 (due on 10/24, Mon)
HW 7 (due on 11/9, Wed) - updated on 11/7: #2 and Section 6.1 #6 are excluded and will be HW 8.
HW 8 (due on 11/16, Wed)
HW 9 (due on 11/30, Wed) - #2: "with 1" should be added.
HW 10 - no need to submit

**Recent Updates and Remarks for Math 519 - Fall 2016**

[Dec 9, 2016] Lecture 43: Chapter 8.1-8.3 - more properties of maximal, prime ideals. / Chapter 9 - definition of polynomial ring, properties, special polynomial rings.

[Dec 7, 2016] Lecture 42: Chapter 8.1-8.3 - more properties of UFD, PID, ED, prime elements, irreducible elements, prime ideal, maximal ideal, examples.

[Dec 5, 2016] Lecture 41: Chapter 8.1-8.3 - examples of UFD, PID, ED, more properties of them and ID.

[Dec 2, 2016] Lecture 40: Chapter 7.6 - application in number theory of chinesse remainder theorem / Chapter 8.1-8.3 - recall def of integral domain, zero divisors, examples, properties in integral domains, introduced UFD, PID, ED.

[Nov 30, 2016] Lecture 39: Chapter 7.5/7.6 - comaximal ideals, chinesse remainder theorem, examples. HW 10 (no need to submit) is handed out.

[Nov 28, 2016] Lecture 38: Chapter 7.4/7.5 - more about kernel, ideals, factor rings, comaximal ideals.

[Nov 21, 2016] Lecture 37: Chapter 7.3/7.4 - ring homomorphism, properties, kernel, properties, subring, ideals.

[Nov 18, 2016] Lecture 36: Chapter 7.1/7.2/7.3 - new terms in rings, examples, categories rings, diagram, examples. Graded HW 8 is returned with solution. HW 9 (due on 11/30, Wed) is handed out.

[Nov 16, 2016] Lecture 35: Chapter 7.1 - plans, outlines for ring theory, definition of rings

[Nov 14, 2016] Lecture 34: Chapter 6.1 - some properties related to nilpotent groups, solvable groups, examples, categories of cyclic < abelian < nilpotent < solvable. Graded HW 7 is returned with solution.

[Nov 9, 2016] Lecture 33: Chapter 6.1 - introduced upper central series, lower central series, derived series, definition of nilpotent group, examples. HW 8 (due on 11/16, Wed) is handed out.

[Nov 7, 2016] Lecture 32: Chapter 5.5 - introduced semi-direct product, properties, examples, classification of group of order 12.

[Nov 4, 2016] Lecture 31: Chapter 4.5 - applications of Sylow 3 theorems. / Chapter 5.2 - def of finitely generated group, fundamental theorem of f.g. abelian groups, Betti numbers, invariant factors, examples, some classification of groups.

[Nov 2, 2016] Lecture 30: Chapter 4.5 - proof of Sylow 3 theorems. Graded EXAM 2 is returned with solution./ HW 7 (due on 11/9, Wed) is handed out.

[Oct 31, 2016] Lecture 29: Chapter 4.5 - examples, proof of Sylow 3 theorems. Graded HW 6 is returned with solution.

[Oct 28, 2016] Lecture 28: Exam 2

[Oct 26, 2016] Lecture 27: Chapter 4.4 / 4.5 - more about automorphism, inner automorphism, examples, stated 3 Sylow thms.

[Oct 24, 2016] Lecture 26: Chapter 4.4 - automorphism, inner automorphism.

[Oct 21, 2016] Lecture 25: Chapter 4.3 - class equation for S_n, and examples, proof of A5 is simple.

[Oct 19, 2016] Lecture 24: Chapter 4.2 / 4.3 - permutation representation, class equation. Graded HW 5 is returned with solution.

[Oct 17, 2016] Lecture 23: Chapter 4.1 / 4.2 - recall the def of group action, some arguments, examples. HW 6 (due on 10/24, Mon) is handed out.

[Oct 14, 2016] Lecture 22: Chapter 4.4 - statement of Sylow theorems, an example.

[Oct 12, 2016] Lecture 21: Chapter 3.4 - introduced solvable groups, examples, several remarks/theorems related to solvable groups. / introduced subjects to be covered in Chapter 4. Graded HW 4 is returned with solution.

[Oct 7, 2016] Lecture 20: Chapter 3.4 - composition series, examples. HW 5 (due on 10/17, Mon) is handed out.

[Oct 5, 2016] Lecture 19: Chapter 3.3 - more details about three isomorphism theorem, examples.

[Oct 3, 2016] Lecture 18: Chapters 3.2 - defined the factor group, examples, why `normal' is required / Chapter 3.3 - stated three isomorphism theorems.

[Sept 30, 2016] Lecture 17: Chapters 3.2 - left/right cosets, partition of G into cosets, equivalence relation, the index [G:H], Lagrange theorem, corollaries. HW 4 (due on 10/7, Fri) is handed out.

[Sept 28, 2016] Lecture 16: Chapters 3.1 - simple groups, example, general arguments related to simple groups.

[Sept 26, 2016] Lecture 15: Chapters 3.1 - group homomorphisms, kernel, normal subgroups, examples. Graded EXAM 1 is returned with solution.

[Sept 23, 2016] Lecture 14: Exam 1

[Sept 21, 2016] Lecture 13: Chapters 2.4 - <A> the subgroup generated by a subset A in a group G, examples / Chapter 2.5 - the lattice of subgroups, examples. Graded HW 3 is returned with solution.

[Sept 19, 2016] Lecture 12: Chapters 2.3 - proved or gave a sketch of proof for the five main properties related to cyclic groups.

[Sept 16, 2016] Lecture 11: Chapters 2.2 - stabilizer, commutator subgroups, their properties, examples. / Chapters 2.3 - def of cyclic group, summary of five main properties related to cyclic groups.

[Sept 14, 2016] Lecture 10: Chapters 2.2 - centralizer, normalizer, center, examples, their properties. Graded HW 2 is returned with solution.

[Sept 12, 2016] Lecture 9: Chapters 2.1 - def of subgroups, examples, arguments related to subgroups, subgroup criterion and proof. HW 3 (due on 9/19, Mon) is handed out.

[Sept 9, 2016] Lecture 8: Chapters 1.7 - more examples of group actions, several new terms of group actions.

[Sept 7, 2016] Lecture 7: Chapters 1.7 - definition, motivation of group actions, examples. Graded HW 1 is returned with solution / HW 2 (due on 9/12, Mon) is handed out.

[Sept 5, 2016] HW 2 (due on 9/12, Mon) is sent out.

[Sept 2, 2016] Lecture 6: Chapters 1.6 - properties of homomorphisms and isomorphisms regarding identity, inverse, subgroup, kernel, order of an element, and etc., monomorphism, embedding, epimorphism, endomorphism, automorphism

[Aug 31, 2016] Lecture 5: Chapters 1.2-1.5 - discussed Q_8 - the quaternion group of order 8 / Chapters 1.6 - gave definitions of homomorphism and isomorphism.

[Aug 29, 2016] Lecture 4: Chapters 1.2-1.5 - an algebraic definition of D_{2n}, group presentation, M_n(F), GL_n(F).

[Aug 26, 2016] Lecture 3: Chapters 1.2-1.5 - cycles, transpositions, even, odd, A_n, a geometric definition of D_{2n}. HW 1(due on 9/2, Fri) is distributed.

[Aug 24, 2016] Lecture 2: more properties of a group, order of an element in a group, order of a group / Chapters 1.2-1.5 - Symmetric group of degree n, permutation,

[Aug 22, 2016] Lecture 1: Syllabus distributed / Chapter 1.1 - binary operation, associative, commutative, definition of groups, examples, properties of groups.

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