Monday, 4/17 notes
Old business:
Group properties:
* order
* generators
* subgroups
* Aut(G)
* conjugacy classe
* abelian
* simple
Smallest simple: A5 (Galois, 1830's)
* discuss its conjugacy classes
* discuss its "presentation" briefly
New business:
Overview of "big" results of group theory
* factoring polynomials => A5 is first non-solvable,
=> general quintic cannot be factored
define: "solvable" group : series of abelian normal "divisors"
define: fields,
full shebang: Galois theory
* classifying groups (look up "simple group" in MathWorld)
(a) factor into simple groups
(b) all simple groups:
(1) cyclic, prime order
(2) An, n>=5
(3) Lie-type Chevally groups (finite field Lie-analog)
(4) Lie type
(5) Sporadic groups
(c) exactly 26 sporadic groups (!),
biggest is the "monster" group
Other directions:
* Lie groups:
- manifold and group
- exponential map e^A where A is an element of tangent space;
- Lie algebra (commutator relations of A's) gives local structure
* Representations of groups => matrices, theoretical physics,