### 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

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

Lecture notes: Groups, Rings, Polynomials

Handouts: Zorn's lemma

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.