A Group

In Group theory:

A group is a non-empty set G together with a binary operation * such that the following four axioms hold:

1. Closure

a * b ∈ G for all a, b ∈ G.

2. Associativity

(a * b) * c = a * (b * c) for all a, b, c ∈ G.

3. Identity element

There exists an element e ∈ G such that e * a = a * e = a for every a ∈ G.

4. Inverse element

For every a ∈ G, there exists a⁻¹ ∈ G such that a * a⁻¹ = a⁻¹ * a = e.

If the operation also satisfies a * b = b * a for all a, b ∈ G, then G is called an Abelian group.