Group Theory

'll throw some light on the title question of this page by asking another question. What is the solution of the equation 

(1) 4x = 3

The answer depends on what "things" we allow x to be. If we are doing all our arithmetic using the integers then there is no solution--there is no integer that gives 3 upon being multiplied by 4. On the other hand if we are doing our arithmetic in Z/5 ("Integers mod 5" as it's sometimes called) then x = 2 is a solution. If we are using the more usual rational number system Q, then the solution is x = 3/4.

We can gain insight into all such questions by considering the equation 

(2) a • x = b

and then bringing up the question of solutions. Well, what objects are a and b? To what class of objects is x allowed to belong? What is the operation symbolized by the dot (•)?

Group theory is concerned with systems in which (2) always has a unique solution. The theory does not concern itself with what a and b actually are nor with what the operation symbolized by • actually is. By taking this abstract approach group theory deals with many mathematical systems at once. Group theory requires only that a mathematical system obey a few simple rules. The theory then seeks to find out properties common to all systems that obey these few rules.

The axioms (basic rules) for a group are:

1. CLOSURE: If a and b are in the group then a • b is also in the group.

2. ASSOCIATIVITY: If a, b and c are in the group then (a • b) • c = a • (b • c).

3. IDENTITY: There is an element e of the group such that for any element a of the group

a • e = e • a = a.

4. INVERSES: For any element a of the group there is an element a-1 such that

* a • a-1 = e 

and

* a-1 • a = e

That's it. Any mathematical system that obeys those four rules is a group. The study of systems that obey these four rules is the basis of GROUP THEORY

A Look at the Axioms

Closure

CLOSURE: If a and b are in the...