Abelian Group

An Abelian group (or commutative group) is a set G equipped with an operation + that satisfies the following properties:

1. Closure: For all a, b ∈ G, a + b ∈ G.
2. Associativity: For all a, b, c ∈ G,

(a + b) + c = a + (b + c).

3. Identity element: There exists 0 ∈ G such that, for every a ∈ G,

a + 0 = 0 + a = a.

4. Inverse element: For every a ∈ G, there exists −a ∈ G such that

a + (−a) = (−a) + a = 0.

5. Commutativity: For all a, b ∈ G,

a + b = b + a.

Therefore, the set of complex numbers under addition, (C, +), is an Abelian group.