Direct Product of Groups - Definition

Definition

Given groups G and H, the direct product G × H is defined as follows:

1. The elements of G × H are ordered pairs (g, h), where g ∈ G and h ∈ H. That is, the set of elements of G × H is the Cartesian product of the sets G and H.
2. The binary operation on G × H is defined componentwise: (g₁, h₁) · (g₂, h₂) = (g₁ · g₂, h₁ · h₂)

The resulting algebraic object satisfies the axioms for a group. Specifically:

Associativity

The binary operation on G × H is indeed associative.

Identity

The direct product has an identity element, namely (1_G, 1_H), where 1_G is the identity element of G and 1_H is the identity element of H.

Inverses

The inverse of an element (g, h) of G × H is the pair (g−1, h−1), where g−1 is the inverse of g in G, and h−1 is the inverse of h in H.