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.
    a-1(ab) = a-1(ac)
    (a-1a)b = (a-1a)c
    b = c
  2. ae = a
    let ae = d
    a-1ae = a-1d
    a-1d = e
    by definition, a-1a = e
    from consequence 1, d = a
  3. e is unique
    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
    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)
    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.