MTWRF 11:30-12:25 CC (Enarson Classroom Building) 312
Instructor: Sasha Leibman
office:
MW (Math Tower) 406
e-mail:
leibman.1@osu.edu
phone: 614-620-7767
Textbook: D.S.Summit and R.M.Foote, Abstract Algebra, 3rd edition
Lecture notes: Groups, Rings, Polynomials
Handouts: Zorn's lemma
Midterm —
solutions
Topics,
review problems
Final exam, due by Tuesday, 12/12.
Homework:
| Homework 1 – | due by Tuesday, August 29. — Solutions. |
| Homework 2 – | due by Wednesday, September 6. — Solutions. |
| Homework 3 – | due by Tuesday, September 12. — Solutions. |
| Homework 4 – | due by Tuesday, September 19. — Solutions. |
| Homework 5 – | due by Tuesday, September 26. — Solutions. |
| Homework 6 – | due by Tuesday, October 3. — Solutions. |
| Homework 7 – | due by Tuesday, October 10. — Solutions. |
| Homework 8 – | due by Wednesday, October 18. — Solutions. |
| Homework 9 – | due by Wednesday, October 25. — Solutions. |
| Homework 10 – | due by Tuesday, November 7. — Solutions. |
| Homework 11 – | due by Tuesday, November 14. — Solutions. |
| Homework 12 – | due by Tuesday, November 21. — Solutions. |
| Homework 13 – | due by Tuesday, December 5. — Solutions. |
Calendar: [LN=Lecture Notes, TB=Text Book]
| August 22: | Binary operations, neutral and inverse elements, semigroups, monoids, groups (LN 1.1, TB 1.1) |
| August 23: |
The cancellation property of groups
(LN 1.3, TB 1.1)
Powers of elements (LN 1.4, TB 1.1) Isomorphic groups. Groups of orders 1, 2, 3, 4 (LN 1.5.4) |
| August 24: |
Groups of orders 6
(LN 1.5.4)
Examples of groups: numbers, residues, matrices (LN 1.2.1-2) |
| August 25: |
Examples of groups: mappings, symmetric groups, symmetry groups,
(LN 1.2.3)
Exercises from Section 1.1 |
| August 28: | Exercises from Section 1.1 |
| August 29: | Examples of groups: set-theoretical groups, fundamental and braid groups, groups of words and free groups (LN 1.2.4-6) |
| August 30: |
The quaternion group Q8
(LN 1.2.7, TB 1.5)
Finite groups of matrices (LN 1.8, TB 1.4) Subgroups (LN 1.6, TB 2.1) |
| August 31: |
Generating sets and relations
(LN 1.7.1-4, TB 2.4)
Presentation of groups in terms of generators and relations (LN 1.7.5-7, TB 1.2) |
| September 1: |
Distinct presentations of a group
(LN 1.7.7)
Finitely generated and cyclic groups (LN 1.7.8, TB 2.4) Exercises from Sections 1.2 |
| September 5: | Exercises from Sections 1.4, 1.6, 2.1, 2.4 |
| September 6: |
Cyclic groups and their subgroups
(LN 1.9, TB 2.3)
The lattice of subgroups of a group (LN 1.10, TB 2.5) The symmetric group Sn, cyclic decomposition of permutations (LN 1.12.1-6, TB 1.3) |
| September 7: |
The symmetric group Sn, parity of permutations,
the alternating group An
(LN 1.12.7-14, TB 1.3)
The center of a group and centralizers of elements (LN 1.13, TB 1.2) |
| September 8: |
Exercises from Sections 1.3, 2.2
Cosets and the index of a subgroup, the 1st counting principle (LN 2.1.1-8, TB 3.1-2) |
| September 11: |
Lagrange's theorem and its corollaries
(LN 2.1.9-13, TB 3.1-2)
The 2nd and 3rd counting principles (LN 2.1.14-16, TB 3.1-2) Normal subgroups (LN 2.2.1, TB 3.1) |
| September 12: |
Conjugation
(LN 2.3.1-3, TB 1.3)
Normal subgroups and factorization (LN 2.2.1-5, TB 3.1) Examples of quotient groups (LN 2.2.6-7, TB 3.1) |
| September 13: |
Normalizers and centralizers of subgroups
(LN 2.3.4-5, TB 2.2, 3.1)
Simple groups. (LN 2.4, TB 3.4) Subnormal and composition series. (LN 2.4, TB 3.4) |
| September 14: | Conjugacy classes in Sn. The simplicity of An for n≥5 (LN 2.5, TB 4.6) |
| September 15: | Exercises from Section 3.1 |
| September 18: |
Exercises from Sections 3.2, 4.6
Homomorphisms of groups (LN 3.1, TB 1.6, 3.1) |
| September 19: |
The kernel of a homomorphism and the 1st isomorphism theorem
(LN 3.2.1-4, TB 3.1, 3.3)
The 2nd isomorphism theorem (LN 3.2.6, TB 3.3) |
| September 20: |
The 3rd isomorphism theorem
(LN 3.2.7, TB 3.3)
Groups as factors of free groups (LN 3.2.5) Reduction of a homomorphism to a quotient group (LN 3.3) The lattice isomorphism theorem (LN 3.2.8, TB 3.3) |
| September 21: |
The Jordan-Hölder theorem
(LN 3.4)
Actions of groups (LN 4.1.1-2, TB 1.7, 4.1) |
| September 22: |
Actions of groups: orbits and stabilizers
(LN 4.1, TB 1.7, 4.1)
Exercises from Section 4.1 |
| September 25: | Exercises from Sections 3.3, 3.4, 1.7 |
| September 26: | The regular left action of groups, Cayley's theorem (LN 4.2, TB 4.2) |
| September 27: |
The action of groups on themselves by conjugations
(LN 4.3, TB 4.3)
Direct products of groups, external and internal (LN 5.1, TB 5.1, 5.4) |
| September 28: | Direct products of groups (LN 5.1.8-10) |
| September 29: | Exercises from Sections 4.2, 4.3 |
| October 2: |
Exercises from Section 5.1
Direct products of several and of inifinitely many groups (LN 5.3, 5.4, TB 5.1) |
| October 3: |
The central and the relative direct products
(LN 5.2)
The Chinese remainder theorem (LN 6.1, TB Proposition 5.6 on p.163) The fundamental theorem of finite abelian groups — existence (LN 6.2.1, TB 5.2) |
| October 4: | Invariant factors and elementary divisors (LN 6.2.2-4, TB 5.2) |
| October 5: |
Invariant factors and elementary divisors
(LN 6.2.2-4, TB 5.2)
The fundamental theorem of finite abelian groups — uniqueness (LN 6.2.5, TB 5.2) Exercises from Section 5.1 |
| October 6: | Groups Zn* (LN 6.3) |
| October 9: |
Exercises from Sections 5.1, 5.2
Groups of automorphisms of groups (LN 7.1.1-3, TB 4.4) |
| October 10: |
Inner and outer automorphisms
(LN 7.1.4-6, TB 4.4)
Characteristic subgroups (LN 7.2, TB 4.4) |
| October 11: | Semidirect products of groups (LN 7.3.1-7, TB 5.5) |
| October 16: | Examples of semidirect products (LN 7.3.8-10, TB 5.5) |
| October 17: |
Isomorphic semidirect products
(LN 7.3.11-13, TB 5.5)
p-groups (LN 8.1, TB 6.1) |
| October 18: |
Groups of orders p2 and p3
(LN 8.3.1-2, TB page183)
Sylow's theorems (LN 8.2.1, TB 4.5) |
| October 19: |
Sylow's theorems
(LN 8.2, TB 4.5)
Groups of order pq (LN 8.3.3, TB 4.5, 5.5) |
| October 20: |
Groups of order pqk, p2q and pqr
(LN 8.3.5-8, TB 4.5, 5.5)
Groups of order 12 (LN 8.2.6, TB 4.5, 5.5) |
| October 23: |
Groups of orders 40 and 24
(LN 8.3.9, TB 4.5, 5.5)
Sylow subgroups of a subgroup and of a quotient group (LN 8.2.8,9) |
| October 24: |
Groups of orders 36 and 48
(LN 8.3.9, TB 4.5, 5.5)
Some methods of proving that a finite group is non-simple (LN 8.3.4, TB 6.2) |
| October 25: |
Commutators and the derived subgroup
(LN 9.1, TB 6.1)
Derived series and solvable groups (LN 9.2, TB 6.1) |
| October 26: |
Central series and nilpotent groups
(LN 9.3, TB 6.1)
Subgroups of free groups (LN 10, TB 6.3) |
| October 27: | Review. |
| October 30: | Midterm. |
| October 31: | Definition and examples of rings (LN (Rings) 1.1, TB 7.1-2) |
| November 1: |
Zero divisors, units, idempotent, nilpotent, unipotent elements
(LN (Rings) 1.2.1, TB 7.1)
Constructions of rings (LN 1.3, TB 7.1-2) |
| November 2: | Fields of fractions (LN 1.4, TB 7.5) |
| November 3: |
Rings of fractions
(LN 1.4, TB 7.5)
Exercises from Sections 7.1,2,5 |
| November 6: |
Ideals
(LN 1.5, TB 7.3)
Homomorphisms of rings (LN 1.6, TB 7.3) |
| November 7: |
Isomorphism theorems for rings
(LN 1.7, TB 7.3)
Direct product of rings (LN 1.8, TB 7.2) |
| November 8: |
Principal ideals
(LN 2.1, TB 7.4)
Divisibility of ideals, gcd and lcm of ideals (LN 2.2, TB 7.4) Comaximal ideals and the Chinese remainder theorem (LN 2.3, TB 7.6) |
| November 9: | Prime and maximal ideals (LN 2.4.1-8, TB 7.4 and Appendix 1) |
| November 13: | Exercises from Sections 7.3,4 |
| November 14: |
Nilradical and Jacobson radical
(LN 2.4.9-13)
A very brief introduction to algebraic geometry (LN 3, TB 15.2) |
| November 15: | Algebraic geometry: units, zero divisors, nilpotent elements, homomorphisms, subrings and quotient rings (LN 3, TB 15.2) |
| November 16: |
Noetherian rings
(LN 2.6.1-7, TB p.316)
The primary decomposition theorem (LN 2.6.8, TB p.684) |
| November 17: |
Prime and irreducible elements of a ring
(LN 4.1, TB 8.3)
Principal ideal domains (LN 4.2, TB 8.2) |
| November 20: |
Unique factorization domains
(LN 4.3, TB 8.3)
Euclidean domains (LN 4.4.1-4, TB 8.1) |
| November 21: |
Euclidean algorithm, universal side divisors
(LN 4.4.5-7, TB 8.1)
The Dedekind-Hasse norm (LN 4.5, TB 8.2) |
| November 27: | Quadratic integer rings (LN 5.1-2, TB 7.1, 8.3) |
| November 28: | Prime ideals and elements in quadratic integer rings (LN 5.3, TB 8.3) |
| November 29: |
Prime elements in Z[i] and representation of positive integers as a sum of two squares
(LN 5.4, TB 8.3)
Polynomials – definition and properties (LN (Polynomials) 1, TB 9.1,2) |
| November 30: |
The polynomial ring over a Noetherian ring is Noetherian
(LN 1, TB 9.6)
Roots of polynomials (LN 2.1-7, TB 9.5) |
| December 1: |
Multiple roots, differentiation
(LN 2.8-11, TB 9.5)
Polynomials over fields (LN 3, TB 9.2,5) Polynomials over UFDs, content (LN 4.1-6, TB 9.3) |
| December 4: | Gauss's lemma, the polynomial ring over a UFD is a UFD (LN 4.7-12, TB 9.3) |
| December 5: |
Criteria of irreducibility of polynomials
(LN 5, TB 9.4)
Exercises from Sections 9.1-3 |
| December 6: | Exercises from Sections 9.3-5 |
Good luck with final, and have a nice winter break.