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
Monday, October 24: Symmetric group and permutations. Cayley's theorem.
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.
Wednesday, November 2: Direct product of groups.
Finitely generated abelian groups. Section 11.
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.
Friday, November 11: More of factor groups. Section 14.
Monday, November 14: Factor group computations and applications.
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
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.
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!