Math 444/544 FALL 2011, List of lectures

• On this page I will post content of all lectures. All handouts also will be posted here.
• Monday, September 26: Proofs, sets, cardinality. Section 0.
• Wednesday, September 28: Relations and equivalence relations. Section 0. Please find your homework.
• Friday, September 30: Examples of groups. Binary operations. Sections 1 and 2.
• Monday, October 3: More on binary operations. Isomorphism of binary structures. Sections 2 and 3.
• Wednesday, October 5: Structural properties and non-isomorphic binary structures. Section 3. New homework.
• Friday, October 7: Definition of a group. Section 4.
• Monday, October 10: Multiplication tables. Groups of order 2,3,4. Parenthese are irrelevant. Powers of an element. Subgroups. Sections 4 and 5. Graduate students: please find here topics for long-term projects.
• Wednesday, October 5: Examples of subgroups. Subgroups diagram. Cyclic subgroups. Section 5. New homework.
• Friday, October 7: Cyclic groups. Subgroups of cyclic groups. Section 6.
• Monday, October 17: Subgroups of finite cyclic groups. Generators of cyclic groups. Subgroups diagram of cyclic group. Section 6. We will have MIDTERM on Friday!!!
• Wednesday, October 19: Automorphisms group of a cyclic group. Subgroup generated by some elements. Generators. Cayley (di)graphs. Symmetric groups and permutations. Sections 6-8. New homework.
• Friday, October 21: MITERM. The solutions are here.
• Monday, October 24: Symmetric group and permutations. Cayley's theorem. Section 8.
• Wednesday, October 26: Cycles and disjoint cycles. Orders of permutations. Section 9. New homework.
• Friday, October 28: Odd and even permutations. Alternating group. Cosets as equivalence classes. Sections 9-10.
• Monday, October 31: Cosets and Lagrange Theorem. Groups of prime order. Section 10.
• Wednesday, November 2: Direct product of groups. Finitely generated abelian groups. Section 11. New homework.
• Friday, November 4: Finite abelian groups. Homomorphisms. Sections 11 and 13.
• Monday, November 7: Image and kernel of a homomorphism. Normal subgroups. Section 13.
• Wednesday, November 9: Factor groups. Section 14. New homework.
• Friday, November 11: More of factor groups. Section 14.
• Monday, November 14: Factor group computations and applications. Section 15. We will have MIDTERM (covering Sections 7-15) next Friday!!! Please find practice problems! Please find solutions.
• Wednesday, November 16: Complements: center, commutant, dihedral groups, homomorphism to the symmetric group of cosets. Sections 15-16. New homework. Please find here class notes (thanks to Jackson Lusk!).
• Friday, November 18: MIDTERM. Please find solutions here.
• Monday, November 21: Rings: definition, examples, homomorphisms and isomorphisms, group of units. Section 18.
• Wednesday, November 23: Multiplicative group of Z_n. Euler's function and its computation. Fields. Field Z_p=F_p. Sections 18-19. New homework.
• Monday, November 28: Zero divisors and integral domains. Fermat's and Euler's theorems. Sections 19-20. Please find practice problems for the final!
• Wednesday, November 30: The field of quotients of an integral domain. Section 21. Last homework.
• Friday, December 2: Review. Please find solutions for practice problems!
• END