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,