What Is a Group in Mathematics?
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. Closure: for all g₁, g₂ ∈ G , the product g₁ * g₂ is still in G . 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. Inverses: for each g ∈ G there exists an element g⁻¹ ∈ G such that g * g⁻¹ = g⁻¹ * g = e . 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 i...