Math319_F18

Introduction to Abstract Algebra I

- Fall 2018 -

Instructor: Kwangho Choiy

• Office: Neckers 283 (618-453-6508)

• E-mail: kchoiy_at_siu_dot_edu 

• Office Hours: Monday/Thursday 12pm - 3pm, or by appointment. Emails are also available for simple questions.

Course Website: https://sites.google.com/site/kchoiy/teaching/math319_f18

Class Meeting: MWF 11:00am - 11:50am in NKRS 156

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], UPDATED on Aug. 14.

Exams: There will be two mid-term exams and one final at the regular classroom NKRS 156. No make-up exam will be accepted.

1.  Exam 1 (in class on 9/24, Mon) covers Sections 0--6.

2.  Exam 2 (in class on 10/29, Mon) covers Sections 8--11, 13, 14

3.  Final Exam (2:45pm-4:45pm on 12/12, Wed) will be comprehensive: approximately, 40% on Sections 0–6, 8–11, 13, 14 and 60% on Sections 15, 16, 18 – 21. - announced on 12/3 in class, by email.

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 8/31, Fri)

2. HW 2 (due on 9/10, Mon)

3. HW 3 (due on 9/17, Mon)

4. HW 4 (due on 10/5, Fri)

5. HW 5 (due on 10/12, Fri)

6. HW 6 (due on 10/19, Fri)

7. HW 7 (due on 11/7, Wed) --**Extended to 11/9, Fri + Delete #2 & Questions Section 16 that will be on HW 8**

8. HW 8+9 (due on 11/28, Wed)

9. HW 10 (no-need-to-submit)

Updates and Remarks - math319 - Fall 2018:

[Dec 7, Fri] Review for Final

[Dec 5, Wed] Review for Final

[Dec 3, Mon] Section 20 - recall two crucial arguments from GT, RT, examples, stated Fermat little theorem, Euler theorem, proofs, examples.  Final Exam (2:45-4:45, 12/12, Wed) is announced in class and by E-mail with Practice Problems.

[Nov 30, Fri] Section 19 - properties of I.D., examples / Section 21 - motive and introduction to Quot(R), the quotient field of integral domains. HW 8+9 is returned with solution.

[Nov 28, Wed] Section 19 - motive of integral domains, defined commutative ring, zero-divisors, integral domains, examples, properties of I.D. HW 8+9 is collected / HW 10 (no-need-to-submit) is distributed.

[Nov 26, Mon] Section 18 - set-group-ring, motives of three axioms examples, cautions, properties, ring homomorphism, direct product of rings, examples.

[Nov 16, Fri] Section 18 - kinds, properties of ring, examples

[Nov 14, Wed] Section 18 - definition of rings, commutative rings, unity, units, division rings, fields, examples Theorem 18.8

[Nov 12, Mon] Section 16 - properties of group actions, bijection between the orbit and the set of left cosets, proof, examples, the partition of X in terms of orbits, |X| = sum of indices. HW 7 is returned with solution 

[Nov 9, Fri] Section 16 - definition of group actions, several examples, new terms, G_x the isotropic subgroup, Gx the orbit of x, examples. HW 7 is collected. / HW 8+9 (due on 11/28, Wed) is distributed.

[Nov 7, Wed] Section 15 - more examples, more explanations on properties of Z(G) and C(G) / Section 16 - two motivations of group actions.

[Nov 5, Mon] Section 15 - D_n, explicit description of D_3, D_4, properties of general D_n, computed Z(D_4), important properties of Z(G), definition of C(G), the commutator subgroup, example, properties of C(G).

[Nov 2, Fri] Section 15 - more examples and remarks for for Z(G). HW 7 due is on 11/9, Fri + Delete #2 & Questions Section 16, as noted above. 

[Oct 31, Wed] Reviewed Exam 1 / Outline of upcoming sections, motivation / Section 15 - definition of Z(G), the center of G, example, remark. Graded Exam 2 is returned with solution / HW 7 (due on 11/7, Wed --> 11/9, Fri) is distributed.

[Oct 29, Mon] Exam 2 is taken.

[Oct 26, Fri] Reviewed for Exam 2.

[Oct 24, Wed] Section 14 - proof of Aut(G) is a group; Inn(G) <| Aut(G).

[Oct 22, Mon] Section 14 - more examples for factor groups, defined and discussed Inn(G) <| Aut(G), example.  HW 6 is returned with solution / Exam 2 (10/29, MON) is announced in class and by E-mail.

[Oct 19, Fri] Section 14 - several properties of factor groups, examples, counterexample for G=S_3, H=<(12)> and (gH)(g'H)=gg'H is not well-defined. HW 6 is collected.

[Oct 17, Wed] Section 14 - proof and remark for the argument, well-definedness of (gH)(g'H)=gg'H iff H <| G.

[Oct 15, Mon] Section 14 - more about normal subgroups, [G:H]=2 => H is normal, three equivalenct arguments, A_n, the alternating group <| S_n, several examples, introduced factor group. Graded HW 5 is returned with solution.

[Oct 12, Fri] Section 13 - subgroup preservance via homomorphism, more properties of homomorphism, left/right cosets of kernel, normal subgroups, examples. HW 6 (due on 10/19, Fri) is distributed / HW 5 is collected.

[Oct 10, Wed] Section 11 - new notations for direct product, direct sum / Section 13 - properties of homomorphism, kernel, examples.

[Oct 8, Mon] Section 10 - applications to Lagrange theorem / Section 11 - classification of finite abelian groups, fundamental theorem for finitely generated abelian groups, examples. Graded HW 4 is returned with solution.

[Oct 5, Fri] Section 10 - further relevant arguments for Lagrange theorem, examples, index, property of index, examples. HW 5 (due on 10/12, Fri) is distributed / HW 4 is collected.

[Oct 3, Wed] Section 10 - Lagrange theorem, its proof, examples, left/right cosets, |H|=|gH|, equivalence relation related to cosets.

[Oct 1, Mon] Section 9 - more discussions on S_n, orbit, permutation in S_n is expressed by a product of disjoint cycles, unique up to order / by a product of transpositions, examples.

[Sep 28, Fri] Section 8&9 - left multiplication lambda_g, right multiplication rho_g, S_n, examples. HW 4 (due on 10/5, Fri) is distributed.

[Sep 26, Wed] Section 8 - proved Caley theorem, examples / reviewed Exam 1. Graded Exam 1 is returned with solution.

[Sep 24, Mon] Exam 1 is taken.

[Sep 21, Fri] Section 8 - permutations, group of permutations, expression of permutation, cycle, length of cycle, transposition, examples / Reviewed for Exam 1.

[Sep 19, Wed] Section 8 - motivating example of permutations, diagram of cyclic, abelian, non-abelian with important examples, the direct product of groups, order of group, element. Graded HW3 is returned with solution.

[Sep 17, Mon] Section 6 - examples for subgroup of cyclic group, properties of finite cyclic groups, proof of the classification theorem of cyclic groups. HW 3 is collected / Exam 1 (9/14, MON) is announced in class and by E-mail.

[Sep 14, Fri] Section 6 - listed properties of cyclic groups, their proofs.

[Sep 12, Wed] Section 6 - motivating example of cyclic groups, cyclic subgroups, <g> is really a subgroup, generator, examples. Graded HW2 is returned with solution.

[Sep 10, Mon] Section 5 - image of inverse of a via iso is inverse of the image of a, motivating argument of subgroup, definition of subgroup, examples, remarks, an important criterion for subgroups, example of isomorphisms, strategies to prove/disprove a statement for isomorphism, identity element, its two important properties. HW 3 (due on 9/17, Mon) is distributed / HW 2 is collected.

[Sep 7, Fri] Section 4 - more examples for groups, properties of groups and proofs, examples

[Sep 5, Wed] Section 3 - proof of image of (identity) via isomorphism is the identity, examples / Section 4 - motivating example, definition of the inverse element, definition of group, examples. Graded HW1 is returned with solution.

[Aug 31, Fri] Section 3 - example of isomorphisms, strategies to prove/disprove a statement for isomorphism, identity element, its two important properties.  HW 2 (due on 9/10, Mon) is distributed / HW 1 is collected.

[Aug 29, Wed] Section 2 - Cayley table / Section 3 - injective, surjective, bijective functions, homomorphism, isomorphism, examples

[Aug 27, Mon] Section 2 - function in terms of relation, binary operation, algebraic structure, examples, commutative, associative.

[Aug 24, Fri] Section 0 - cardinality, def of same cardinality, aleph zero, one, example / Section 1 - Euler formula, solution in the complex number, U_n and Z_n. HW 1 (due on 8/31, Fri) is distributed

[Aug 22, Wed] Section 0 - relation, equivalence relation, partition, cell, examples, theorem stating equivalence relation <=> partition.

[Aug 20, Mon] Syllabus distributed in class. Main objectives of the course. Section 0 - brief concepts groups, numbers, sets

[Aug 14, Tue] Supplementary Material: Visit the link to see mathematical proofs, notation, arguments, etc.