Math 5590H, Algebra I

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

Syllabus

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

Lecture notes: Groups, Rings, Polynomials

Handouts: Zorn's lemma

Midtermsolutions
Topics, review problems

Final examsolutions

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.