Introduction to Group Theory

The axioms (basic rules) for a group G are:

1. CLOSURE: Ifthenwhereis the rule for composing elements of the group.

2. ASSOCIATIVITY: Ifthen

3. IDENTITY: There is an elementsuch that for any element a of the group
   The identity is unique.

4. INVERSES: For any elementof the group there is an unique elementsuch that

Closure

Ifthenis also in the group.

For a system to be a group the group operation must hold for any pair of elements in the group and the result of the operation must be an element of the group. The set of negative integers, for example, is not closed under multiplication because the product of two negative integers is not a negative integer.

Associativity

Ifthen

Addition and multiplication of real numbers is associative but not subtraction and division. For examplebut

Thus

Identity

There is an elementof the group such that for any elementof the group

Cross product of three dimensional vectors is an example of a binary operation that does not have an identity element. Since the cross product ofwith any other vector is either the zero vector or a vector perpendicular to there can be no vectorwith (otherwisewould be perpendicular to itself).

Inverses

For any elementof the group there is an elementsuch that

In order for an operation to satisfy the axiom for inverses the operation must have an identity element. Any vector does not have an inverse under the cross product operation since there is no identity.