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The direct product G × H has elements (g, h) with operation (g₁, h₁)(g₂, h₂) = (g₁g₂, h₁h₂). G × H is abelian iff both factors are. The order is |G| · |H| and direct products are commutative: G × H ≅ H × G.
The direct product is the simplest way to build a new group out of two groups you already know. Recall from your study of groups that a group needs a set, an operation, an identity, inverses, and associativity. The direct product G × H satisfies all of these by running G and H simultaneously and independently: elements are ordered pairs (g, h), and the operation works component-wise — the G-components multiply together using G's operation, and the H-components multiply using H's. Neither component knows or cares about the other.
A concrete example makes this tangible. Take G = ℤ₂ = {0, 1} under addition mod 2, and H = ℤ₃ = {0, 1, 2} under addition mod 3. Then G × H has 6 elements: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2). Adding (1,2) + (1,1) gives (1+1 mod 2, 2+1 mod 3) = (0,0). The identity is (0,0) and every element has an inverse. In fact, ℤ₂ × ℤ₃ ≅ ℤ₆ — these are the same group in disguise, because gcd(2,3) = 1. This is the Chinese Remainder Theorem for groups: when the orders are coprime, the direct product of cyclic groups is again cyclic.
The abelian property follows immediately from the component-wise definition. If both G and H are abelian, then (g₁,h₁)(g₂,h₂) = (g₁g₂, h₁h₂) = (g₂g₁, h₂h₁) = (g₂,h₂)(g₁,h₁). Conversely, if G × H is abelian but G is not, you can find non-commuting elements in G and lift them to non-commuting pairs in G × H, reaching a contradiction. The order formula |G × H| = |G| · |H| reflects the fact that you're forming all possible pairings — a Cartesian product in the set sense — so counting is just multiplication.
Direct products are also the natural decomposition tool for finite abelian groups. The Classification Theorem for Finite Abelian Groups — which you will meet soon — states that every finite abelian group is isomorphic to a direct product of cyclic groups of prime-power order. Understanding direct products is therefore not just about combining two groups; it is about recognizing when a group you encounter in the wild is secretly built from simpler pieces. Whenever you can write G ≅ H₁ × H₂ × ⋯ × Hₖ, you have decomposed a complex structure into independently operating components, each of which you can analyze separately.