Direct product of groups

The identity of \(G\times H\) is \((e_G,e_H)\) and the inverse of \((g,h)\) is \((g^{-1},h^{-1})\); associativity is inherited from each factor. If \(G\) and \(H\) are finite, \(|G\times H|=|G|\,|H|\). For example, in \(\mathbb{Z}_6\times\mathbb{Z}_4\) the sum \((2,3)+(5,2)\) is done in each coordinate with its own modulus and gives \((1,1)\).

The product is abelian exactly when both factors are. Moreover, the order of an element is computed with the least common multiple: \(\operatorname{ord}(g,h)=\operatorname{lcm}(\operatorname{ord}g,\operatorname{ord}h)\). This yields a useful criterion: \(\mathbb{Z}_m\times\mathbb{Z}_n\) is cyclic if and only if \(\gcd(m,n)=1\), and in that case it is isomorphic to \(\mathbb{Z}_{mn}\). So \(\mathbb{Z}_3\times\mathbb{Z}_8\cong\mathbb{Z}_{24}\), but \(\mathbb{Z}_2\times\mathbb{Z}_2\) is not cyclic: all its nontrivial elements have order 2.

Each factor appears inside the product as the subgroup \(G\times\{e_H\}\) (and similarly \(\{e_G\}\times H\)), and the projections \((g,h)\mapsto g\) and \((g,h)\mapsto h\) are surjective homomorphisms. These subgroups are normal, commute with each other and intersect trivially, which allows us to recognize when a given group «is» a direct product.

A common mistake is to believe that \(\mathbb{Z}_m\times\mathbb{Z}_n\) is always isomorphic to \(\mathbb{Z}_{mn}\): that happens only for coprime moduli. Another is to confuse the direct product with the Cartesian product of sets, which has no operation. Later, the structure theorem will express every finite abelian group as a direct product of cyclic groups.