What Is a Group in Mathematics?

In abstract algebra, a group is a set G together with a binary operation (written as *). The pair (G, *) is called a group when the operation satisfies the four conditions below.

1. Closure: for all g₁, g₂ ∈ G, the product g₁ * g₂ is still in G.
2. Identity: there exists an element e ∈ G such that for all g ∈ G, e * g = g * e = g. This element e is called the identity.
3. Inverses: for each g ∈ G there exists an element g⁻¹ ∈ G such that g * g⁻¹ = g⁻¹ * g = e.
4. Associativity: for all g₁, g₂, g₃ ∈ G, g₁ * (g₂ * g₃) = (g₁ * g₂) * g₃.

These four conditions are exactly what is required for (G, *) to be a group.

Side note: Commutativity

An operation is commutative if swapping the elements does not change the result: a * b = b * a. Commutativity is not required for a group to exist.

It is important not to confuse commutativity with associativity. These are distinct ideas:

Property	Statement	Required for a group?
Associativity	(a * b) * c = a * (b * c)	Yes
Commutativity	a * b = b * a	No

What makes a group Abelian?

A group is called Abelian (or commutative) if its operation satisfies a * b = b * a for all elements in the group. In other words, an Abelian group is simply a group where the order of combination does not matter.

So, every Abelian group is a group, but not every group is Abelian.

Examples

• Abelian: integers under addition (ℤ, +) — all four axioms hold, and addition is commutative.
• Non-Abelian: square matrices (size ≥ 2) under multiplication — the group axioms hold, but matrix multiplication is not commutative, since AB ≠ BA in general.