| Week |
Session |
Topics |
Chapter |
Prerequisite Skills |
Learning Outcomes |
Examples |
Exercises
|
| 1
|
Z
|
- The natural order on N and the well ordering principle
- Mathematical induction
- Construction of Z and its properties (graph the equivalence classes)
- Division algorithm
- Congruence mod m
- Algebra on the quotient set Z_m
- GCD, LCM, Bézout
- Primes, Euclid's Lemma
- Fundamental Theorem of Arithmetic
|
0
|
- Sets
- Partitions
- Equivalence relations and classes
- Functions
- Images and preimages
|
- Review of known facts about Z
- A concrete introduction to techniques of abstract algebra
|
- Equivalence classes are partitions
- If f is a function, xRy <=> f(x)=f(y) is an equivalence relation and the equivalence classes are fibers of f.
- Introduce congruence using the remainder function.
- Congruence classes mod 3
- Extended Euclid's algorithm
|
|
| 2
|
Groups
|
- Symmetries
- Properties of composition
- Definition of a group
- Elementary proofs with groups:
- uniqueness of identity
- uniqueness of inverses
- cancellation
- shortcuts to establishing group axioms
- Foundational examples with Cayley tables
|
2
|
Sets and functions
|
- Motivation for the concept of a group
- Learn the definition of a group
- Learn basic automatic properties of groups (with proofs) for later use as shortcuts
- Starting to build a catalog of examples of groups
- Learn to construct and read Cayley tables
|
- Z, Q, Q*, Q+, R, R*, R+, {-1, 1}
- R^n, M(n,R)
- symmetric group S_n
- Z_2 defined for now as {even,odd} ({solids,stripes})
- correspondence of Z_2 with {-1, 1} and with S_2
- functions X -> G with pointwise operation (fg)(x)=f(x)g(x)
- free group on a finite set
|
|
| 3
|
Homomorphisms
|
- Cayley's theorem
- Homomorphisms of groups
- Isomorphisms and their inverses
- Automorphisms
- Examples
|
10, 6
|
- Functions
- Groups
- Matrix multiplication
- Change of basis for matrices
|
- General framework for thinking of groups as symmetries and motivation for homomorphisms
- Learn the definitions of homomorphism and isomorphism
- Prove that homomorphisms preserve powers.
- Starting to build a catalog of examples of homomorphisms.
|
- R -> R: x -> ax
- R^n -> R^n: v -> Av
- M(n,R) -> M(n,R): X -> AX
- R* -> R*: x -> x^n
- R -> R+: x -> a^x (a>0)
- determinant: GL(n,R) -> R*
- inclusions
- natural projection Z -> Z_2
- evaluation {X -> G} -> G: f -> f(a)
- Z -> Z: k -> -k
- Aut(Z_2) is trivial
- Aut(Z_3) is isomorphic to Z_2
- change of basis S in R^n gives an inner automorphism of GL(n,R): X -> S^(-1).X.S
- C -> C: z -> complex conjugate of z
|
|
| 4
|
Subgroups
|
- Definition of a subgroup
- Subgroup tests
- Automatic closure under inverses for finite subgroups
- Subgroups generated by a subset
- Examples
- Images and preimages under a homomorphism are subgroups.
- Fibers as cosets of the kernel
- First Isomorphism Theorem
- Examples
|
3, 10
|
- Groups
- Functions
- Equivalence relations and classes
|
- Learn how to identify subgroups, with proofs.
- Learn how to obtain new groups from old via homomorphisms.
- Learn how to prove a homomorphism is one-to-one by using the kernel.
|
- Cyclic subgroups <x>={x^k: k in Z} or xZ={xk: k in Z}
|
|
| 5
|
Groups in Linear Algebra and Complex Variable
|
- Euclidean space as an additive group
- Null space and column space of a linear map
- Solutions to linear inhomogeneous systems
- Invertible linear transformations and matrices, GL(n,R)
- Determinant: homomorphism, similarity invariance, geometrical interpretation.
- Additive and multiplicative subgroups of complex numbers
|
|
|
|
- GL(n,R), O(n,R), SL(n,R), SO(n,R)
- C, C*, S^1, n-th roots of unity, D_n as a subgroup of O(2,R^2)
- Möbius transformations on the Riemann sphere
|
|
| 6
|
Cyclic groups
|
- Order of a group, order of an element
- Defining homomorphisms on Z (free group)
- Classification of cyclic groups
- Subgroups of cyclic groups and their generators
- Subgroup lattice
|
4
|
|
|
|
|
| 7
|
Permutations
|
- Cycle notation
- D_n as a subgroup of S_n
- Factoring into disjoint cycles
- Ruffini's theorem
- Cyclic subgroups, powers of a permutation
- Parity, A_n < S_n
|
5
|
|
|
|
|
| 8
|
- Catch up and review
- Midterm 1
|
| 9
|
Cosets
|
- Cosets as equivalence classes
- Lagrange's theorem
- Fermat's little theorem
- Euler's theorem
- Normal subgroups
- Factor groups
- Universal property of factor groups
- First Isomorphism theorem revisited
|
7, 9
|
|
|
- cosets of <(1,2)> in S_3
- cosets of a flip in D_4
- inverse images of subgroups are normal, kernels
- A_n is normal in S_n
- rotations in D_n
- Z/nZ
- R/Z
|
|
| 10
|
Products
|
- External direct product
- Universal property of direct product
- Chinese Remainder Theorem
- Internal direct product
- Free product
- Universal property of free product
- Direct sums of Abelian groups and classification of finitely generated Abelian groups (without proof)
|
8, 9, 11
|
|
|
- ZxZ
- public key cryptography
- coproduct of Z with itself in groups and in Abelian groups
- free group on a set
- free Abelian group on a set
|
|
| 11
|
Rings
|
- Motivation and definition
- Properties
- Subrings
- Integral domains
- Fields
- Characteristic
- Ring homomorphisms
- Examples
|
12, 13, 15, 16
|
|
|
- Z and other number systems
- R*
- Z_n* = U(n)
- polynomial rings
|
|
| 12
|
Ideals and factor rings
|
- Ideals
- Ideals generated by a set, principal ideals
- Images and preimages of ideals are ideals
- Factor rings
- Prime ideals
- Maximal ideals
- Localization, field of quotients
|
14
|
|
|
- mZ < Z
- <2, x> = 2Z[x]+xZ[x] < Z[x]
- Hausdorff Maximality Principle
- Q[x]/<x^2-2>
- R[x]/<x^2+1>
- Z -> Q
- polynomials -> rational functions
|
|
| 13
|
Factorization
|
- Division algorithm for F[x]
- F[x] is a PID
- Factorization of polynomials
- Fundamental Theorem of Algebra
- Tests, Eisenstein's criterion
- Irreducibles and associates
- Z[x] is a UFD
|
16, 17, 18
|
|
|
- In Z[x]/<x^2+5> we have 6=2.3=(1-sqrt(-5))(1+sqrt(-5))
|
|
| 14
|
- Catch up and review
- Midterm 2
|
| 15
|
- Catch up and review for final
- Study days
|
