Disclaimer: although the plan is to have a fairly detailed list of the covered topics, the list below might not cover everything that we discussed in class.
Book: David Dummit & Richard Foote. Abstract Algebra. Wiley. 3th ed.
Week 1
Lecture 1, 09/03. Introduction; binary operations; definition of a group (based on D&F 1.1).
Syllabus
Binary operations
Definition of a group
First examples, , , , ,
Abelian groups
Example of non-abelian group,
Lecture 2, 09/05. Equivalence classes and (based on D&F 0.3).
I suppose you know 0.1 and 0.2 of D&F
Equivalence relations and equivalence classes
as a set
operations on and their correctness
is a group
Example:
Week 2
Lecture 3, 09/08. Group properties and more about (based on D&F 0.3, 1.1).
Identity unique
Inverse unique
Long associativity
Multiplicative notation
Solving linear equations
Order of an element and order of a group
Examples in
is not a group
Lecture 4, 09/10. Dihedral group (based on D&F 1.2).
Dihedral group of symmetries on regular -gon
Elements of
maps are associative
and as generators
Lecture 5, 09/12. Symmetric group (based on D&F 1.2, 1.3).
Dihedral group as linear functions on
Symmetric group of a set
is a group of permutations of
how to write and work with permutations
-cycles
every permutation is a product of independent cycles
Week 3
Lecture 6, 09/15. (based on D&F 1.3, 0.3).
Example: , permutations, cycle notation, order
Example: large , permutation, cycle notation, order
Example: product of dependent cycles
Examples: ,
Examples: ,
Lecture 7, 09/17. Matric groups and Quaternion groups (based on D&F 1.4, 1.5).
Definition a Field
Fields: , , ,
Linear algebra over any field
, 6 elements
Matrix representation of
Lecture 8, 09/19. Homomorphisms and isomorphisms (based on D&F 1.6).
Definition of homomorphism
Examples: everything to 1; ; ;
Epimorphism, monomorphism
Isomorphism, isomorphic groups
Examples: ;
Isomorphism preserve all group properties: orders, abelian
Week 4
Lecture 9, 09/22. Group Action (based on D&F 1.7).
Classification theorems
explicitly
Group action .
Examples
is a bijection
Permutation representation: homomorphism
Faithfull actions
Lecture 10, 09/24. Subgroups, Center, Centralizer, Normalizer (based on D&F 2.1, 2.2).
Definition of Subgroup
Subgroup Criterion
Examples
Which conditions can be omitted
The Center of a group
Examples
The Centralizer of a set in a group
Normalizer of a set in a group
Properties,
Lecture 11, 09/26. Homomorphism’s Image and Kernel, Action’s Stabilizer and Kernel (based on D&F 2.2).
Example: ,
Kernel and Image of a homomorphism
Stabilizer of an element of an action
Kernel of an action
Examples
Centralizer, Normalizer as action’s kernel
Week 5
Lecture 12, 09/29. Cyclic groups I (based on D&F 2.3).
Cyclic groups, generators of cyclic groups
Cyclic subgroups
Examples and nonexamples: , , , , , , ,
Lecture 13, 10/01. Cyclic groups II (based on D&F 2.3).
if and , then .
if , then
All cyclic groups ar isomorphic to or ()
Subgroups of cyclic groups are cyclic groups with classification of such subgroups
Lecture 14, 10/03. Subgroups generated by subset of a group (based on D&F 2.4).
as
as
section 2.5 for self reading
Week 6
Lecture 15, 10/06. Quotient Group (based on D&F 3.1).
section 2.5 for self reading
Review of homomorphisms
Factor group of a homomorphism as a group of fibers
Left and right cosets by subgroup
Examples
Lecture 16, 10/08. Review (based on D&F 0.1–2.3).
Lecture 17, 10/10. Normal Subgroups (based on D&F 3.1).
Fibers are left (right) cosets by kernel
Factor group of a homomorphism as a group of left cosets
Normal subgroups
Factor group by a normal subgroup
Examples: , .
Week 7
Lecture 18, 10/13. Cosets and Lagrange Theorem (based on D&F 3.2).
Cosets is a partition of a group
Index of a subgroup
Examples
Lagrange Theorem
The order of the element is a divisor of the order of the group
Groups of order .
Morning section: about groups of order
Afternoon section: subgroups of index 2 are normal.
Lecture 19, 10/15. First and Third Isomorphisms Theorems (based on D&F 3.3).
Criterium of normal subgroup: for any
Kernel is normal subgroup
First Isomorphisms Theorem:
Corollaries
Third Isomorphisms Theorem: .
Lecture 20, 10/17. Fourth and Second Isomorphisms Theorems (based on D&F 3.2, 3.3).
Example of 3rd Theorem
Fourth Isomorphisms Theorem: Bijection between lattices
Example of 4th Theorem
as a set
Second Isomorphisms Theorem:
Week 8
Lecture 21, 10/20. Alternating group (based on D&F 3.5).
Review of our knowledge about
Cycle as a product of transpositions
as a homomorphism
Even and odd permutations? Alternating group
Inversions of permutations
as a parity of number of inversions
For matrix we have
Lecture 22, 10/22. Group Actions (based on D&F 4.1).