GROUPS (under construction)

Definition from axioms.

A group consists of a set G and a binary operation * with the following properties:

• Associativity
  
• Closure
  
• Identities
  
• Inverses
  

Definition: A binary operation * is commutative if x*y = y*x for all values of x and y.

Theorem: The identity element of a group is unique.      Proof.

Theorem: The inverse of an element of a group is unique.   Proof.

Theorem:     Proof.

Examples of groups.

Modulo Arithmetic.

Matrix Groups.

Permutation Groups.

Symmetry groups

• Equilateral Triangle - D₃,
• Square,
• Regular Pentagon,
• Regular Hexagon,
• Non-Square Rectangle (Klein four group),
• Regular Tetrahedron.