Simple groups, Lie groups, and the search for symmetry

Math Online Tom Circle

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)

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