## Introduction to Group Theory

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

1. CLOSURE: If then where is the rule for composing elements of the group.

2. ASSOCIATIVITY: If then 3. IDENTITY: There is an element such that for any element a of the group The identity is unique.

4. INVERSES: For any element of the group there is an unique element such that Closure

If then is 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

If then Addition and multiplication of real numbers is associative but not subtraction and division. For example but Thus Identity

There is an element of the group such that for any element of 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 of with any other vector is either the zero vector or a vector perpendicular to there can be no vector with (otherwise would be perpendicular to itself).

Inverses

For any element of the group there is an element such 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.