16A. Direct products and Classification of Finite Abelian Groups

Definition. Let G and H be groups. Their direct product is the group G×H defined as follows. As a set G×H = {(g, h) : g ∈ G, h ∈ H} is just the usual Cartesian product of G and H (the set of ordered pairs where the first component lies in G and the second component lies in H). The group operation on G×H is defined by the formula (g1, h1)(g2, h2) = (g1g2, h1h2) for all g1, g2 ∈ G and h1, h2 ∈ H. Here g1g2 is the product of g1 and g2 in G and h1h2 is the product of h1 and h2 in H. Verification of group axioms for G ×H is straightforward. The identity element of G × H is the pair (eG, eH) where eG is the identity element of G and eH is the identity element of H. Inverses in G×H are given by the formula (g, h)−1 = (g−1, h−1).