* Intuition of "a group":  D4
                       12
  diamond clock      9    3
                       6
  How many transformations back onto same diamond shape?
  * count 'em
  * table of successive transformations - a "binary operation". (Cayley Table)
  * Supposed only allowed to flip top-bottom, and rotate to right. 
     Can you get all the other transformations?  How do you find the right way?
  * Are there any other groups with eight elements?
     (And how can we tell if two groups are the "same", anyway?)

* Mathematics: 
     - formal systems and "proof" : definitions, postulates, rules -> theorems

* Formal definition of a group

* Group with 1 element

* Group with 2 elements

* Cyclic groups, Cn (integers, mod n, under addition)

* Is there another group with 4 elements?

================ didn't do this stuff yet

* definition: "abelian".     (" a*b=b*a ")
* definition: "isomorphic."  ("_really_ the same")

* definition:  "cycle" of an element
* definition:  "order" of an element

* patterns in how many groups?

* solid cube group :   heading towards thinking about Rubik's Cube