Group Theory - A Non-mathematic Introduction

1. The word Group does mildly relate to its everyday usage as in a group of things, elements, people, actions, operations, 'groups of objects', etc.
2. To make a group of things into a Mathematic Group we need to define a binary operation on any 2 members of the group. This operation (or method) needs to have 4 special properties. So needless to say, an ordinary group of things is most likely not a Group since the required operation does not exist.
3. In Rubik's Cube, a move can be single slice-turn or a long sequence of many turns. On this group of move we can combine 2 moves to form a longer one (i.e. first move followed by second). This method conforms to
   property #1(Closure Law): two elements in the group when combined by some method results in an element which also belongs to the group.
   Here, the group of things is the group of all possible moves. The method or operation of combining 2 moves is simply "make the first move then make the second move". The result is also a move.
4. The operation used is really up to us and our ingenuity. Example1: A group of 10 numbers from 0 to 9. We can combined 2 numbers by adding them. But this will violate property #1 since combining 7 and 8 results in 15 which is not part of the group. However if we cleverly decree that after combining if the result is more than 10 we subtract 10 from the result. This new method will have property #1 since the result of combining 7 and 8 is 5. If we take 'close' to mean 'confined to the group', then this is the so-called Closure Law: combination of any 2 Group elements also belong to the Group.
5. Any Rubik's Cube move sequence has a magic-number which is the number of repeats to return to the original state.

   From reset state ( ) the move when repeated 6 times returns to the reset state. Magic-number (actually known as order) of the move is 6.

   If we denote original state by I, RightTopX by a, RightTopX twice as a², 3-times as a³, etc., then

   {a,a²,a³, a⁴,a⁵,I } forms Cyclic6 Group with 6 elements.
   {a²,a⁴,a⁶=I } a subgroup of 3 elements (of order 3),
   {a³,a⁶=I} a subgroup of order 2; a³ interestingly swaps 2 pairs of corners, so to swap back, just repeat a³.

   Fact: Any Rubik's Cube move sequence, no matter how simple or complex generates a cyclic group. Examples: Order of = 105; order of = 72. Try doing any 3-turn and determine its magic-number by repeating the same 3-turn until done.

6. When combining 3 things we can combine first two then combine the result to third or we can combine last two then combine the first with the result. For Cyclic6 group (in fact any cyclic group) it does not matter! For example,
   (a²+a⁵)+a³= a²+(a⁵+a³) =doing RightTopX 10 times
   =a⁴=RightTopX 4 times because magic-number is 6.

   This is

   property #2(Associative Law): When combining 3 elements by the specified method (or operation), the order of combining does not matter: (a op b) op c = a op (b op c).

   This associative property says that as long as we combine left to right, we are free to factor or bracket any 2 elements. So a+b+c has a clear meaning. Not so if a method is non-associative since (ab)c may be different from a(bc). A common non-associative example is subtraction: (5-3)-2 is not the same as 5-(3-2).

   Example2 non-associative: group of 10 numbers from 0 to 9

   The method of combining 2 numbers is to take the absolute difference. Here

   (3 combine with 2) combine with 1 = (1) - 1 = 0
   3 combine with (2 combine with 1) = 3 - (1) = 2
   which shows that the method is not associative.

   
   Example3 non-assocative: group of orientations of square paper

     A+-----+B
      |     |
      |     | 
     D+-----+C	  
   

   A 3D orientation is a rotation or a flip. The result of combining 2 orientations is that obtained by doing the first then the second orientation. If we identify the 4 corners as ABCD clockwise from left top, then rotate 90° clockwise results in DABC; flip left edge to right edge results in BADC.

   (rotate combine with rotate) combine with flip = CDAB+flip = DCBA
   rotate combine (rotate combine with flip) = rotate+ADCB = BADC
   which shows that the method is not associative.
7. In Cyclic6 Group, we have a "no move" move sequence identified by a⁶ or I. This element plays an important role in Group Theory:
   property #3(Identity law): There is a special element called the identity such that it combines with any element a to give a: a+I=a.
   We need this identity element to tell us that a combination of many move sequences is equivalent to no-move such as a²+a⁴=I. and it is also needed to define what is called an inverse in property #4.
8. The Cyclic6 Group has a nice property that we can combine any of the 6 moves with just one element to result in the identity. Example: a²+a⁴=I. This is
   property #4(Inverse law): For every element, we can find another element called its inverse such that the combination is the identity.
   Obviously, the inverse of a² is a⁴ and vice versa.
9. Definition non-mathematical: A Group is defined by a set of things and a method of combining 2 of those things that conforms to 4 special properties.

   Definition mathematical: A Group is a set of elements with a binary operation which is close and associative. It has an identity element and each element has an inverse.

10. What is the big deal about Group? Stay tuned...