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.
• 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).
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.