group theory (idea)

Group theory is often introduced a bit imprecisely. Perhaps this is because its practitioners are too familiar and comfortable with it. I will introduce a concise definition of a group and will prove a few important consequences of the definition.

A group is a set of elements, G, with a mapping G x G --> G (i.e. all pairs of elements in G are mapped to an element of G). The mapping of an element pair (a,b) to an element c is indicated by the expression ab = c. The mapping, called the group product, must obey the following rules:

1. a(bc) = (ab)c (associative property)
2. An element e exists such that for all a in G, ea = a.
3. For each element a in G, an element b exists such that ba = e. b is denoted by a^-1.

It is important to realize that ab is not necessarily equal to ba. If ab = ba for all elements a and b, then the group is termed abelian.

That's all! This definition of a group is a bit more concise than the earlier definitions. The extras in those definitions can be proven as consequences of this definition.

Consequences of this definition:

1. ab = ac implies b = c.
   Proof:
   a^-1(ab) = a^-1(ac)
   (a^-1a)b = (a^-1a)c
   b = c
   
2. ae = a
   Proof:
   let ae = d
   a^-1ae = a^-1d
   a^-1d = e
   by definition, a^-1a = e
   from consequence 1, d = a
3. e is unique
   Proof:
   suppose fa = a
   f = f
   f = fe (because of consequence 2)
   f = e
   or suppose af = a
   f = f
   f = ef
   f = e
4. aa^-1 = e
   Proof:
   let aa^-1 = d
   a^-1aa^-1 = a^-1d
   a^-1 = a^-1d
   from consequence 2, a^-1e = a^-1
   from consequence 1, d = e
5. ba = ca implies b = c (addendum to consequence 1)
   Proof:
   baa^-1 = caa^-1
   from consequence 4, b = c

An interesting example of a group is the 4-group (Viergruppe) V defined with the following group product. This group is abelian (the group product is commutative). Actually, all groups of order 5 or smaller are abelian.

       e    a    b    c 
      ------------------
   e | e    a    b    c
     |
   a | a    e    c    b
     |
   b | b    c    e    a
     |
   c | c    b    a    e

One could imagine that a represents 180-degree rotation about the x-axis, b represents 180-degree rotation about the y-axis, c represents 180-degree rotation about the z-axis, e represents no rotation, and the group product means that one operation is followed by another. You can perform these rotations with a book in your hand to visualize them. Rotations about different axes are in general not commutative. You can verify this by performing 90-degree rotations with the book.