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 (1G, 1H), where 1G is the identity element of G and 1H 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.

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