Abelian group

https://en.wikipedia.org/wiki/Abelian_group

Abelian group or commutative group is a group where the axiom of commutativity is also satisfied. Abelian groups generalize the arithmetic of addition of integers.

Abelian group: the Set of Satisfied Axioms (SSA)

• CLO Closure (Totality)

• IDE Identity

• INV Invertibility

• ASS Associativity

• COM Commutativity

Abelian group is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative.

With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples.

The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras.

The theory of abelian groups is generally simpler than that of non-abelian groups; finite abelian groups are very well understood and fully classified.

Non-abelian group

https://en.wikipedia.org/wiki/Non-abelian_group

A non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there are at least two elements a and b that are non commutative with respect to the group operation: a ∗ b ≠ b ∗ a

∃a,∃b ∈ G . a ∗ b ≠ b ∗ a

This class of groups contrasts with abelian groups which are commutative (all group elements commute).

Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.