Math 5590H and 5111, Algebra I

MWF 11:30-12:25 BO128, TR 11:30-12:25 HA012

Instructor: Sasha Leibman office: MW406     office hours: Wednesday 2-3pm, Thursday 1-2pm
e-mail: leibman.1@math.osu.edu
phone: 614-292-0663

Syllabus

Textbook: D.S.Summit and R.M.Foote, Abstract Algebra, 3rd edition

Midterm 1 — September 28. Solutions. List of topics. Sample problems.
Midterm 2 — November 1. Solutions. List of topics. Sample problems.
There will be no Midterm 3, sorry.
The final exam scheduled days are December 11 and December 14, of your choice.

Homework:

HW1 recommended: by Friday, August 25 -- Exercises 1.1.1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 27, 28, 29, 32, 33, 36 (pages 21-23).
written part: due by Tuesday, August 29 -- Exercises 1.1.9, 22, 25, 30, 31 (pages 22-23).

HW2 recommended: by Friday, September 1 -- Exercises 1.2.3, 4, 5, 6, 8, 9–13, 14, 15 (page 28);
1.3.1, 4, 5, 6, 7, 8, 9, 13, 16, 18 (pages 32-34);
2.1.3, 4, 7, 8, 9, 11, 12, 14, 15 (pages 48-49);
2.3.1, 2, 3, 13, 16 (pages 60-61);
2.4.1, 2, 8, 9, 10, 11, 13, 14, 15, 19, 20 (pages 65-66).
written part: due by Tuesday, September 5 -- Exercises 1.2.7 (page 28); 1.3.15 (page 33); 1.5.3 (page 36); 2.1.6 (page 48); 2.3.12 (page 60); 2.4.6, 7 (page 65).

HW3 recommended: by Friday, September 8 -- Exercises 2.3.25 (page 61);
3.1.3, 4, 5, 15, 16, 17, 22, 23, 24, 25, 27, 28, 29, 30, 31 (pages 85-89);
3.2.1, 2, 3, 6, 11, 12, 18, 20, 23 (pages 95-96).
3.5.2, 3, 4, 7 (page 111).
written part: due by Tuesday, September 12 -- Exercises 3.1.14(a,b,c,d), 34(a,b), 42 (pages 88-89); 3.2.5(a,b), 8, 10 (pages 95-96), 3.5.5 (page 111).

HW4 recommended: by Friday, September 15 -- Exercises 1.6.1, 2, 4, 9, 11, 15, 16, 17, 19, 22 (pages 39-41);
3.1.6, 7, 8, 10, 12, 32, 37 (pages 85-86);
3.3.7, 8, 10 (page 101).
written part: due by Tuesday, September 19 -- Exercises 1.6.18 (page 40); 3.1.9, 33, 36, 38 (pages 85-88); 3.3.3, 4 (page 101).

HW5 recommended: by Friday, September 22 -- Exercises 1.7.1, 2, 3, 8, 9, 10, 12, 14, 15 (pages 44-45);
4.1.4, 5, 6, 10 (pages 116-117);
4.2.1, 3, 7, 9, 10, 11 (page 121-122).
4.3.2, 6, 7, 8, 9, 11, 13, 19, 20, 21, 22, 24, 25, 27, 30, 31, 32, 33, 36 (page 130-132).
written part: due by Tuesday, September 26 -- Exercises 1.7.23 (page 45); 4.1.3, 9(a,b) (page 117); 4.2.4, 6, 8 (pages 121-122); 4.3.34 (page 132).

HW6 recommended: by Friday, October 6 -- Exercises 3.4.1, 7, 9, 10 (page 106);
5.1.1, 2, 3, 7, 15, 17, 18 (pages 156-158);
5.2.1, 2, 3, 4, 5, 7, 9, 10, 13, 14 (page 165-167).
written part: due by Tuesday, October 10 -- Exercises 3.4.2 (the Q8 part only), 5 (page 106); 5.1.14 (page 157); 5.2.1(a,b), 4(b) (pages 165-166);
and two more problems: (1) Prove that any p-group is solvable (where p is a prime);
(2) If groups A and B have relatively prime orders, prove that any subgroup of A×B has the form H×K where H≤A and K≤B.

HW7 recommended: by Monday, October 16 -- Exercises 4.4.1, 6, 7, 12, 17, 18-19 (pages 137-139).
written part: due by Tuesday, October 17 -- Exercises 4.4.5, 8(a,b,c), 13 (pages 137-138).

HW8 recommended: by Friday, October 20 -- Exercises 4.5.1, 2, 4, 6, 9, 13, 16, 18, 19, 24, 26, 29, 30, 32, 33, 34, 36 (pages 146-148);
5.5.1, 2, 4, 5, 14, 18 (pages 184-186).
written part: due by Tuesday, October 24 -- Exercises 4.5.7, 17, 35, 37(a,b,c,d) (pages 146-148); 5.5.6, 9, 13 (pages 184-186).

HW9 recommended: by Tuesday, November 7 -- Exercises 7.1.1-22 (pages 230-232);
7.2.1-9, 12 (pages 237-239).
written part: due by Wednesday, November 8 -- Exercises 7.1.14(a-d), 15, 21(a,b) (pages 231-232); 7.2.2, 10(a,c) (pages 238-239).

HW10 recommended: by Tuesday, November 14 -- Exercises 7.3.1-16, 34, 35 (pages 247-250);
7.4.1-13, 14, 16, 18, 19, 23, 24, 26, 27 (pages 256-259).
written part: due by Wednesday, November 15 -- Exercises 7.3.17, 20, 22(a,b), 33(a,b) (pages 248-250); 7.4.15(a,b,c), 17, 33(a,b,c) (pages 257-259).

HW11 recommended: by Monday, November 20 -- Exercises 8.2.1, 2, 3, 6, 7, 8 (pages 282-283).
written part: due by Tuesday, November 21 -- Exercises 8.2.5(a,b,c) (page 283); 8.3.8(a,b,c) (page 293).

No HW for Thanksgiving

My solutions to HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10, HW11

Studied topics:
August 22: Definition and examples of groups (Sections 1.1 and 0.3)
August 23: Examples of groups (Sections 1.1, 1.2)
August 24: Elementary properties of groups (Section 1.1)
Order of elements in a group (Section 1.1)
Direct products of groups (Example 6 on p.18)
August 25: Exercises from Section 1.1
August 28: Classification of groups of orders 1–6.
Subgroups. Subgroups generated by subsets (Sections 2.1, 2.4).
August 29: Cyclic groups and their subgroups (Section 2.3)
Groups defined by generators and relations (Section 1.2)
August 30: The dihedral group D2n (Section 1.2)
The quaternion group Q8 (Section 1.5)
Finite matrix groups (Section 1.4)
August 31: The symmetric group Sn. Parity of permutations (Sections 1.3 and 3.5)
September 1: Exercises from Sections 1.2, 2.4.
September 5: Subgroups and generators of finite cyclic groups. The groups Zn* (Section 0.3)
The alternating group An (Section 3.5)
September 6: Cosets. Index of a subgroup. (Sections 3.1, 3.2)
Lagrange's theorem and corollaries (Section 3.1)
Counting principles (Section 3.2)
September 7: Normal subgroups (Section 3.1)
Factorization, quotient groups (Section 3.1)
September 8: Groups, defined by generators and relations - revisited.
September 11: Conjugation (Section 4.3)
Properties of normal subgroups (Section 3.1)
The normalizer and the centralizer of a subgroup (Sections 2.2 and 3.1)
Homomorphisms of groups (Section 1.6)
September 12: Properties of homomorphisms (3.1)
The kernel of a homomorphism. Isomorphisms (Section 3.1)
September 13: The 1st isomorphism theorem (Section 3.3)
Representation of any group as a factor of a free group.
Reduction of a homomorphism to a quotient group (page 100).
September 14: The 2nd and the 3rd isomorphism theorems (Section 3.3)
The lattice of subgroups of a group. The 4th isomorphism theorem (Section 3.3)
September 15: Exercises from Sections 1.6, 3.1, 3.3.
September 18: Group actions (Sections 1.7, 4.1)
September 19: Orbits and stabilizers of a group action (Sections 1.7, 4.1)
Actions of a group on itself – by left multiplication (Section 4.2)
Cayley's theorem (Section 4.2)
September 20: Right group actions (Section 4.3)
Actions of a group on itself – by conjugations. Conjugacy classes. Normalizers and centralizers (Section 4.3)
September 21: Conjugacy classes in Sn (Section 4.3)
Simple groups (Section 4.3)
September 22: Exercises from Sections 1.7, 4.1, 4.2, 4.3.
September 25: Exercises from Section 4.3.
The simplicity of An for n≥5 (Section 4.3)
September 26: Review.
September 27: Review.
September 28: Midterm 1.

September 29: Subnormal and composition series. Jordan-Hölder theorem. Hölder's program (Section 3.4)
October 2: Solvable groups (Section 3.4)
Direct products of groups. Internal direct products of subgroups (Sections 5.1, 5.4)
October 3: Internal direct products of subgroups (Sections 5.1, 5.4)
The Chinese remainder theorem (Proposition 5.6 on p.163)
October 4: Direct products and sums of infinite collections of groups (Section 5.1)
The fundamental theorem of finite abelian groups (Section 5.2)
October 5: The fundamental theorem of finite abelian groups (Section 5.2)
October 6: Exercises from Section 5.2.
October 9: Exercises from Sections 3.4, 5.1.
Groups of automorphisms of groups (Section 4.4)
The groups of automorphisms of Q8, Zn, D2n (Section 4.4)
October 10: Inner and outer automorphisms (Section 4.4)
October 11: Groups of automorphisms of Sn, Zpn (Section 4.4)
Characteristic subgroups (Section 4.4)
Semidirect products of groups (Section 5.5)
October 16: Internal and external semidirect products (Section 5.5)
October 17: Examples of external semidirect products (Section 5.5)
Sylow's theorems (Section 4.5)
October 18: Sylow's theorems (Section 4.5)
October 19: Examples of applications of Sylow's theorems (Section 4.5)
Groups of orders pq, pqr (Sections 5.3, 5.5)
October 20: Exercises from Section 4.5.
October 23: Exercises from Sections 4.5, 5.5
October 24: Some problems...
October 25: The group of rotations of a cube.
Groups of orders pqk, p2q (Sections 5.3, 5.5)
October 26: Commutator calculus, derived groups (Sections 5.4, 6.1)
The derived series and solvable groups (Section 6.1)
October 27: The upper and the lower central series. Nilpotent groups (Section 6.1)
October 30: Finite nilpotent groups (Section 6.1)
Subgroups of a free group (Section 6.3)
October 31: Review.
November 1: Midterm 2.

November 2: Rings — definition and examples (Sections 7.1, 7.2)
Elementary properties of rings (Section 7.1)
Zero divisors and units in a ring (Section 7.1)
November 3: Quaternions, Boolean rings, rings of endomorphisms (Section 7.2)
Finite rings without zero divisors (cf. Section 7.1)
Integral domains and their characteristic (Section 7.1)
Fields of fractions (Section 7.5)
November 6: Universality of the field of fractions (Section 7.5)
Rings of fractions (Section 7.5)
Quotient rings and ideals (Section 7.3)
November 7: Examples of ideals.
Exercises from Section 7.1.
November 8: Homomorphisms of rings (Section 7.3)
Isomorphisms of rings. Isomorphism theorems for rings (Section 7.3)
November 9: Properties of ideals (Section 7.4)
Ideals and fields (Section 7.4)
Ideals, generated by sets. Principal ideals (Section 7.4)
November 13: "Divisibility" of ideals. gcd and lcm of two ideals (Section 7.4)
Maximal and prime ideals (Section 7.4)
November 14: Properties of prime ideals (Section 7.4)
Existence of maximal ideals via Zorn's lemma (Section 7.4 and Appendix I)
The nilradical is the intersection of prime ideals (Exercise 26 on p.258)
November 15: Radical ideals (Exercises 30 and 31 on p.258)
Primary ideals (Exercise 41 on p.260)
Noetherian rings (p.316)
November 16: The ring of polynomials over a Noetherian ring is Noetherian (Hilbert's Basis Theorem, p.316)
The primary decomposition theorem (p.684)
Prime and irreducible elements of an integral domain (Section 8.3)
November 17: Rings of quadratic integers (Sections 7.1, p.229, and 8.3)
Unique factorization domains (Section 8.3)
November 20: Principal ideal domains (Section 8.2) and Unique factorization domains (Section 8.3)
November 21: Euclidean domains (Section 8.1)

Plans for the nearest future:
The Dedekind-Hasse norm (Section 8.2)
Primes in the ring of Gaussian integers Z[i] (Section 8.3)