More actions; September 17
This material is covered in Chapter 5 and sections 6.1, 6.2
1
Vocabularycontinued
1. Orbit
2. Transitive
3. Isotropy subgroup of a point.
4. The relationship between isotropy subgroups of two points that lie in the same orbit.
5
Linear Equations, Matrices, and algebraic systems of various sorts;
intro to groups
September 3, 2009
1
Systems of Linear Equations:
This material is covered in Artins Chapter 1, sections 1-3
It is hard to exaggerate the importance of the great success th
Conjugacy, Class Equations, Examples
(Computations in GL2 and the Symmetric Groups)
September 22
This material is (more than) covered in sections 6.2. 6.6, 5.7, 5.8, 4.4
1
Vocabulary Review
Orbit, Transitive, Stabilizer (synonym: Isotropy subgroup) of a p
More Presentations; Homework due October 15; Intro to rings
October 8
Much of this material is covered in sections 6.7. 6.8, 10.1-10.4
1
Presentations
Let x1 , x2 , . . . , xn be n symbols and F := Fn the free group on those generators. So, the set of
ele
Intro to rings, modules, ideals
October 13
This material is covered in sections 10.1-10.4
1
Rings
Denition of rings: recall the conventions, a ring being an associative ring. Discuss rings with
unity, ring homomorphisms (preserving unity; or not). Side co
Groups; isomorphisms; examples; more examples
September 8, 2009
1
Groups
This material is covered in Artins Chapter 2, section 1
A group G is a set with a composition law, i.e., a mapping m : G G G (for any two elements
x, y G, we think of m(x, y) as the
The Symmetric Groups; Quotient groups
September 24
This material is (more than) covered in sections 6.2. 6.6, 5.7, 5.8, 4.4
1
The group Sn
Cycles of length , Transpositions.
Disjoint cycles commute.
Any element Sn is essentially uniquely expressible as
Permutations and actions
(September 15)
1
Reading assignment
Read section 4 of Chapter 1 of Artin (pp. 24-28). The material well do in class today is covered
in 5.8 and 6.1.
2
Permutation Groups
Denition 1 A permutation of a set X is a one:one corresponde
More Quotient groups; Products; . . .
September 29
1
Quotient Groups
1. Denition (Section 2.10)
2. First Isomorphism Theorem Recall
3. Even permutations versus odd permutations. The alternating group An Sn . Quotient of
this.
(Sections 1.4, 6.6)
4. SLn (R
Continuation of Intro to rings, modules, ideals
October 15
This material is covered in sections 10.1-10.4
1
Ideals
Recall denition of ideal (left, right, two-sided) and of quotient rings.
Recall:
Exercise 1 (not to be handed in)
ideal.
The intersection