## Definition of a Group

A group **G** is a set equipped with an associative operation on two elements a, b of **G** that will result in a third element c of **G**.
A group has an invers for each element and a neutral element for the operation.

### Definition

• A group is a pair (**G**, "○" ) consisting of a set **G** and an operation "○".

• The operation a○b -> c maps each pair of elements (a,b), with a, b ∈ **G** to a third element c ∈ **G**.

• The set **G** shall be equipped with a neutral element **e **∈ **G** with the property: **e**○a = a○**e** = a.

• For each element their shall be an invers element a^{-1} ∈ **G** with the properties a○a^{-1} = a^{-1}○a = **e**.

• Associative Property of the operation "○": (a○b)○c = a○(b○c).

### Example

The set ℤ of the integer numbers and the operation "+" form a group (ℤ "+" ).

The operation a+b -> c; for all a,b ∈ ℤ will have the result c ∈ ℤ.

a+0 = 0+a = a; for all a ∈ ℤ; with 0 as the neutral element of ℤ for the operation "+".

a+(-a) = (-a)+a = 0; The element (-a) is the invers of a with (-a), a ∈ ℤ.

(a+b)+c = a+(b+c); with a, b, c ∈ ℤ.

### Commutative Group - Abelian Group

An Abelian group is a group, where the group operation "○" is commutative.

a ○ b = b ○ a for all a,b ∈ **G**;

If the group operation is commutative for all elements of the group set, the group is called Abelian.

### Non Commutative Group - Non-Abelian Group

The definition of a group does not require, that the operation "○" is commutative.

It is possible that a ○ b ≠ b ○ a for some elements a, b ∈ **G**;

A group where the group operation is not commutative is called a non-Abelian group or non commutative group.