Simple groups, Lie groups, and the search for symmetry

Math History | NJ Wildberger

Summary:

1. Normal subgroup
2. Quotient Group
3. Finite Simple Group: Z5, D5,…
4. Tesselation (regular)
5. Platonic solids

… Two families of Group :

1. Symmetric Groups: Sn
2. Linear Group: GL(n) = M, det M ≠0

…

An = Alternating Group

[even permutation, det = +1]

n>=5, An is a Simple (non-commutative) Group

Lie Groups :

SO(2)

SO(3)

Simple Lie Groups : Matrix Group M

1. Special Linear Group: SL(n), det M = 1
2. Orthogonal Group: O(n), $latex M. M^T = I$
3. Symplectic Group : Sp(2n)

…